LMFDB - Abstract group 4769856.a: $\SOPlus(4,13)$

Show commands: Gap / Magma / Oscar / SageMath

comment:Construction of abstract group

magma:G := SOPlus(4,13);

gap:G := Group( (1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440), (1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495), (1,2,8,38,173,301,692,590,238,89,308,577,754,758,726,513,683,309,99,261,60,12,58,252,602,203,601,362,711,717,474,327,421,386,101,21,4,18,87,346,727,742,619,298,321,80,319,492,147,491,317,79,16,3,13,63,151,32,149,495,365,93,363,584,500,611,206,45,204,315,113,139,478,581,775,529,266,546,540,713,579,545,172,37,170,538,548,174,456,600,768,571,221,91,358,738,721,702,384,100,184,176,552,284,685,434,271,316,554,251,180,39,49,193,42,191,487,477,585,703,712,318,326,614,465,776,570,185,40,182,458,129,379,153,438,289,550,511,691,700,303,424,763,777,522,674,265,672,751,693,767,668,257,115,255,392,195,314,646,760,615,208,294,285,311,400,625,628,361,697,426,313,643,678,533,451,730,352,433,118,25,5)(6,28,130,188,136,97,74,71,292,425,663,369,389,745,437,119,62,264,245,55,11,53,114,24,111,416,380,443,360,689,391,747,396,104,22,20,19,92,359,739,446,650,681,378,98,377,488,665,719,757,676,267,659,607,273,453,503,382,504,250,194,256,169,61,35,7,33,154,142,483,460,535,734,355,521,653,243,489,561,179,534,375,160,519,556,279,385,333,225,236,201,397,715,344,86,342,291,161,34,158,338,698,589,782,746,609,627,222,626,226,50,223,383,258,616,572,343,662,634,254,664,413,399,137,29,135,472,569,666,667,608,205,415,112,419,756,411,576,728,390,102,388,673,445,124,420,152,275,65,214,47,213,228,51,10,48,216,622,484,300,351,305,76,304,287,69,277,224,329,401,260,73,70,14,67,280,479,501,642,244,141,30)(9,43,196,587,404,694,490,146,31,144,428,209,117,430,233,52,231,418,190,68,283,543,171,356,90,354,732,733,417,480,744,455,127,27,126,450,373,544,297,272,64,270,435,595,473,568,759,737,675,679,394,631,710,724,341,541,647,240,219,367,720,699,741,575,494,189,41,187,563,701,617,211,507,248,56,181,107,403,687,286,686,761,755,409,109,23,106,215,599,364,422,486,368,95,247,657,374,594,603,530,515,157,514,779,578,671,263,562,640,564,520,574,253,166,134,186,357,736,469,644,237,506,636,432,573,498,150,497,290,637,229,635,340,459,429,116,353,731,604,512,156,509,439,766,591,269,531,296,528,164,527,621,337,84,17,82,328,345,128,427,764,618,705,307,454,629,262,623,752,690,288,688,783,592,197,381,743,649,586,336,210,46)(15,72,145,339,536,278,66,276,448,532,684,630,565,680,481,140,96,370,168,78,59,230,639,293,133,468,349,94,57,249,624,218,268,549,605,414,110,412,452,551,175,138,476,740,366,410,505,177,539,350,88,232,85,325,81,323,105,398,235,212,395,324,436,471,192,583,431,716,440,376,402,613,207,612,310,77,120,143,103,393,749,781,547,274,516,648,773,517,658,582,485,167,36,165,372,651,580,334,83,332,542,718,654,312,406,753,598,722,652,295,610,220,537,496,159,518,725,347,525,162,524,330,163,526,708,784,729,748,467,407,387,322,217,423,199,198,593,696,706,772,567,183,566,677,482,466,132,463,281,682,762,597,320,638,282,148,493,641,660,707,499,241,54,239,645,780,553,202,44,200,596,750,774,709,502,695,331,108,405,242,302,75)(26,121,227,510,234,557,769,441,655,246,632,559,335,123)(122,348,371,588,620,765,449,125,447,457,131,461,444,306)(155,299,606,408,656,661,508,778,523,771,704,633,723,475)(178,555,735,464,770,442,670,259,669,470,714,462,560,558) );

sage:G = PermutationGroup(['(1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440)', '(1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495)', '(1,2,8,38,173,301,692,590,238,89,308,577,754,758,726,513,683,309,99,261,60,12,58,252,602,203,601,362,711,717,474,327,421,386,101,21,4,18,87,346,727,742,619,298,321,80,319,492,147,491,317,79,16,3,13,63,151,32,149,495,365,93,363,584,500,611,206,45,204,315,113,139,478,581,775,529,266,546,540,713,579,545,172,37,170,538,548,174,456,600,768,571,221,91,358,738,721,702,384,100,184,176,552,284,685,434,271,316,554,251,180,39,49,193,42,191,487,477,585,703,712,318,326,614,465,776,570,185,40,182,458,129,379,153,438,289,550,511,691,700,303,424,763,777,522,674,265,672,751,693,767,668,257,115,255,392,195,314,646,760,615,208,294,285,311,400,625,628,361,697,426,313,643,678,533,451,730,352,433,118,25,5)(6,28,130,188,136,97,74,71,292,425,663,369,389,745,437,119,62,264,245,55,11,53,114,24,111,416,380,443,360,689,391,747,396,104,22,20,19,92,359,739,446,650,681,378,98,377,488,665,719,757,676,267,659,607,273,453,503,382,504,250,194,256,169,61,35,7,33,154,142,483,460,535,734,355,521,653,243,489,561,179,534,375,160,519,556,279,385,333,225,236,201,397,715,344,86,342,291,161,34,158,338,698,589,782,746,609,627,222,626,226,50,223,383,258,616,572,343,662,634,254,664,413,399,137,29,135,472,569,666,667,608,205,415,112,419,756,411,576,728,390,102,388,673,445,124,420,152,275,65,214,47,213,228,51,10,48,216,622,484,300,351,305,76,304,287,69,277,224,329,401,260,73,70,14,67,280,479,501,642,244,141,30)(9,43,196,587,404,694,490,146,31,144,428,209,117,430,233,52,231,418,190,68,283,543,171,356,90,354,732,733,417,480,744,455,127,27,126,450,373,544,297,272,64,270,435,595,473,568,759,737,675,679,394,631,710,724,341,541,647,240,219,367,720,699,741,575,494,189,41,187,563,701,617,211,507,248,56,181,107,403,687,286,686,761,755,409,109,23,106,215,599,364,422,486,368,95,247,657,374,594,603,530,515,157,514,779,578,671,263,562,640,564,520,574,253,166,134,186,357,736,469,644,237,506,636,432,573,498,150,497,290,637,229,635,340,459,429,116,353,731,604,512,156,509,439,766,591,269,531,296,528,164,527,621,337,84,17,82,328,345,128,427,764,618,705,307,454,629,262,623,752,690,288,688,783,592,197,381,743,649,586,336,210,46)(15,72,145,339,536,278,66,276,448,532,684,630,565,680,481,140,96,370,168,78,59,230,639,293,133,468,349,94,57,249,624,218,268,549,605,414,110,412,452,551,175,138,476,740,366,410,505,177,539,350,88,232,85,325,81,323,105,398,235,212,395,324,436,471,192,583,431,716,440,376,402,613,207,612,310,77,120,143,103,393,749,781,547,274,516,648,773,517,658,582,485,167,36,165,372,651,580,334,83,332,542,718,654,312,406,753,598,722,652,295,610,220,537,496,159,518,725,347,525,162,524,330,163,526,708,784,729,748,467,407,387,322,217,423,199,198,593,696,706,772,567,183,566,677,482,466,132,463,281,682,762,597,320,638,282,148,493,641,660,707,499,241,54,239,645,780,553,202,44,200,596,750,774,709,502,695,331,108,405,242,302,75)(26,121,227,510,234,557,769,441,655,246,632,559,335,123)(122,348,371,588,620,765,449,125,447,457,131,461,444,306)(155,299,606,408,656,661,508,778,523,771,704,633,723,475)(178,555,735,464,770,442,670,259,669,470,714,462,560,558)'])

sage_gap:G = gap.new('Group( 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)')

oscar:G = @permutation_group(784, (1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440), (1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495), (1,2,8,38,173,301,692,590,238,89,308,577,754,758,726,513,683,309,99,261,60,12,58,252,602,203,601,362,711,717,474,327,421,386,101,21,4,18,87,346,727,742,619,298,321,80,319,492,147,491,317,79,16,3,13,63,151,32,149,495,365,93,363,584,500,611,206,45,204,315,113,139,478,581,775,529,266,546,540,713,579,545,172,37,170,538,548,174,456,600,768,571,221,91,358,738,721,702,384,100,184,176,552,284,685,434,271,316,554,251,180,39,49,193,42,191,487,477,585,703,712,318,326,614,465,776,570,185,40,182,458,129,379,153,438,289,550,511,691,700,303,424,763,777,522,674,265,672,751,693,767,668,257,115,255,392,195,314,646,760,615,208,294,285,311,400,625,628,361,697,426,313,643,678,533,451,730,352,433,118,25,5)(6,28,130,188,136,97,74,71,292,425,663,369,389,745,437,119,62,264,245,55,11,53,114,24,111,416,380,443,360,689,391,747,396,104,22,20,19,92,359,739,446,650,681,378,98,377,488,665,719,757,676,267,659,607,273,453,503,382,504,250,194,256,169,61,35,7,33,154,142,483,460,535,734,355,521,653,243,489,561,179,534,375,160,519,556,279,385,333,225,236,201,397,715,344,86,342,291,161,34,158,338,698,589,782,746,609,627,222,626,226,50,223,383,258,616,572,343,662,634,254,664,413,399,137,29,135,472,569,666,667,608,205,415,112,419,756,411,576,728,390,102,388,673,445,124,420,152,275,65,214,47,213,228,51,10,48,216,622,484,300,351,305,76,304,287,69,277,224,329,401,260,73,70,14,67,280,479,501,642,244,141,30)(9,43,196,587,404,694,490,146,31,144,428,209,117,430,233,52,231,418,190,68,283,543,171,356,90,354,732,733,417,480,744,455,127,27,126,450,373,544,297,272,64,270,435,595,473,568,759,737,675,679,394,631,710,724,341,541,647,240,219,367,720,699,741,575,494,189,41,187,563,701,617,211,507,248,56,181,107,403,687,286,686,761,755,409,109,23,106,215,599,364,422,486,368,95,247,657,374,594,603,530,515,157,514,779,578,671,263,562,640,564,520,574,253,166,134,186,357,736,469,644,237,506,636,432,573,498,150,497,290,637,229,635,340,459,429,116,353,731,604,512,156,509,439,766,591,269,531,296,528,164,527,621,337,84,17,82,328,345,128,427,764,618,705,307,454,629,262,623,752,690,288,688,783,592,197,381,743,649,586,336,210,46)(15,72,145,339,536,278,66,276,448,532,684,630,565,680,481,140,96,370,168,78,59,230,639,293,133,468,349,94,57,249,624,218,268,549,605,414,110,412,452,551,175,138,476,740,366,410,505,177,539,350,88,232,85,325,81,323,105,398,235,212,395,324,436,471,192,583,431,716,440,376,402,613,207,612,310,77,120,143,103,393,749,781,547,274,516,648,773,517,658,582,485,167,36,165,372,651,580,334,83,332,542,718,654,312,406,753,598,722,652,295,610,220,537,496,159,518,725,347,525,162,524,330,163,526,708,784,729,748,467,407,387,322,217,423,199,198,593,696,706,772,567,183,566,677,482,466,132,463,281,682,762,597,320,638,282,148,493,641,660,707,499,241,54,239,645,780,553,202,44,200,596,750,774,709,502,695,331,108,405,242,302,75)(26,121,227,510,234,557,769,441,655,246,632,559,335,123)(122,348,371,588,620,765,449,125,447,457,131,461,444,306)(155,299,606,408,656,661,508,778,523,771,704,633,723,475)(178,555,735,464,770,442,670,259,669,470,714,462,560,558))

Group information

Description:	$\SOPlus(4,13)$
Order:	$4769856$$\medspace = 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order() oscar:order(G)
Exponent:	$2184$$\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent() oscar:exponent(G)
Automorphism group:	$C_2\times \PSL(2,13)^2.D_4$, of order $19079424$$\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage:libgap(G).AutomorphismGroup() sage_gap:G.AutomorphismGroup() oscar:automorphism_group(G)
Composition factors:	$C_2$ x 2, $\PSL(2,13)$ x 2	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries() oscar:composition_series(G)
Derived length:	$1$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage:libgap(G).DerivedLength() sage_gap:G.DerivedLength() oscar:derived_length(G)

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

oscar:is_abelian(G)

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

oscar:is_cyclic(G)

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

oscar:is_nilpotent(G)

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

oscar:is_solvable(G)

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

oscar:is_supersolvable(G)

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage:G.is_simple()

sage_gap:G.IsSimpleGroup()

oscar:is_simple(G)

Group statistics

comment:Compute statistics for the group G

magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;

gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");

sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))

Order	1	2	3	4	6	7	8	12	13	14	21	24	26	28	39	42	52	56	78	84	91	156	168	182
Elements	1	28731	33488	66612	165984	219960	56784	729456	28560	804024	170352	113568	28560	170352	61152	170352	61152	340704	61152	340704	157248	122304	681408	157248	4769856
Conjugacy classes	1	3	3	4	7	15	2	26	4	39	6	4	4	6	2	6	2	12	2	12	6	4	24	6	200
Divisions	1	3	3	4	7	5	2	14	4	13	2	2	4	2	2	2	2	2	2	2	2	2	2	2	86
Autjugacy classes	1	3	2	2	5	9	1	10	2	18	3	2	2	3	1	3	1	3	1	6	3	2	6	3	92

comment:Compute statistics about the characters of G

magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);

gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);

sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]

sage_gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i G.CharacterDegrees()

oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)

Dimension	1	12	13	14	28	36	72	98	144	156	168	169	182	196	336	364	392	432	468	504	1008	2016
Irr. complex chars.	2	12	4	10	0	0	2	2	42	12	72	2	10	30	0	0	0	0	0	0	0	0	200
Irr. rational chars.	2	0	4	6	2	4	2	2	0	0	2	2	6	10	2	2	10	14	4	6	4	2	86

Minimal presentations

Permutation degree:	$784$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	72	72	72
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / Oscar / SageMath

Groups of Lie type:	$\SOPlus(4,13)$
magma:G := SOPlus(4,13); gap:G := Group([[[ 0Z(13), 0Z(13), Z(13)^0, Z(13)^6 ], [ 0Z(13), 0Z(13), 0Z(13), Z(13)^6 ], [ Z(13)^6, Z(13)^6, Z(13)^9, Z(13)^3 ], [ 0Z(13), Z(13)^0, 0Z(13), Z(13)^9 ]], [[ Z(13)^0, 0Z(13), 0Z(13), 0Z(13) ], [ 0Z(13), Z(13), 0Z(13), 0Z(13) ], [ 0Z(13), 0Z(13), Z(13)^11, 0Z(13) ], [ 0Z(13), 0Z(13), 0Z(13), Z(13)^0 ]], [[ 0Z(13), Z(13)^6, 0Z(13), Z(13)^6 ], [ Z(13)^0, Z(13)^9, Z(13)^6, Z(13)^9 ], [ 0Z(13), 0Z(13), 0Z(13), Z(13)^0 ], [ 0Z(13), 0Z(13), Z(13)^6, Z(13)^9 ]]]); sage:MS = MatrixSpace(GF(13), 4, 4) G = MatrixGroup([MS([[0, 0, 1, 12], [0, 0, 0, 12], [12, 12, 5, 8], [0, 1, 0, 5]]), MS([[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 7, 0], [0, 0, 0, 1]]), MS([[0, 12, 0, 12], [1, 5, 12, 5], [0, 0, 0, 1], [0, 0, 12, 5]])]) oscar:G = matrix_group([matrix(GF(13), [[0, 0, 1, 12], [0, 0, 0, 12], [12, 12, 5, 8], [0, 1, 0, 5]]), matrix(GF(13), [[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 7, 0], [0, 0, 0, 1]]), matrix(GF(13), [[0, 12, 0, 12], [1, 5, 12, 5], [0, 0, 0, 1], [0, 0, 12, 5]])])
Permutation group:	Degree $784$ $\langle(1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6) \!\cdots\! \rangle$ (1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440), (1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495), (1,2,8,38,173,301,692,590,238,89,308,577,754,758,726,513,683,309,99,261,60,12,58,252,602,203,601,362,711,717,474,327,421,386,101,21,4,18,87,346,727,742,619,298,321,80,319,492,147,491,317,79,16,3,13,63,151,32,149,495,365,93,363,584,500,611,206,45,204,315,113,139,478,581,775,529,266,546,540,713,579,545,172,37,170,538,548,174,456,600,768,571,221,91,358,738,721,702,384,100,184,176,552,284,685,434,271,316,554,251,180,39,49,193,42,191,487,477,585,703,712,318,326,614,465,776,570,185,40,182,458,129,379,153,438,289,550,511,691,700,303,424,763,777,522,674,265,672,751,693,767,668,257,115,255,392,195,314,646,760,615,208,294,285,311,400,625,628,361,697,426,313,643,678,533,451,730,352,433,118,25,5)(6,28,130,188,136,97,74,71,292,425,663,369,389,745,437,119,62,264,245,55,11,53,114,24,111,416,380,443,360,689,391,747,396,104,22,20,19,92,359,739,446,650,681,378,98,377,488,665,719,757,676,267,659,607,273,453,503,382,504,250,194,256,169,61,35,7,33,154,142,483,460,535,734,355,521,653,243,489,561,179,534,375,160,519,556,279,385,333,225,236,201,397,715,344,86,342,291,161,34,158,338,698,589,782,746,609,627,222,626,226,50,223,383,258,616,572,343,662,634,254,664,413,399,137,29,135,472,569,666,667,608,205,415,112,419,756,411,576,728,390,102,388,673,445,124,420,152,275,65,214,47,213,228,51,10,48,216,622,484,300,351,305,76,304,287,69,277,224,329,401,260,73,70,14,67,280,479,501,642,244,141,30)(9,43,196,587,404,694,490,146,31,144,428,209,117,430,233,52,231,418,190,68,283,543,171,356,90,354,732,733,417,480,744,455,127,27,126,450,373,544,297,272,64,270,435,595,473,568,759,737,675,679,394,631,710,724,341,541,647,240,219,367,720,699,741,575,494,189,41,187,563,701,617,211,507,248,56,181,107,403,687,286,686,761,755,409,109,23,106,215,599,364,422,486,368,95,247,657,374,594,603,530,515,157,514,779,578,671,263,562,640,564,520,574,253,166,134,186,357,736,469,644,237,506,636,432,573,498,150,497,290,637,229,635,340,459,429,116,353,731,604,512,156,509,439,766,591,269,531,296,528,164,527,621,337,84,17,82,328,345,128,427,764,618,705,307,454,629,262,623,752,690,288,688,783,592,197,381,743,649,586,336,210,46)(15,72,145,339,536,278,66,276,448,532,684,630,565,680,481,140,96,370,168,78,59,230,639,293,133,468,349,94,57,249,624,218,268,549,605,414,110,412,452,551,175,138,476,740,366,410,505,177,539,350,88,232,85,325,81,323,105,398,235,212,395,324,436,471,192,583,431,716,440,376,402,613,207,612,310,77,120,143,103,393,749,781,547,274,516,648,773,517,658,582,485,167,36,165,372,651,580,334,83,332,542,718,654,312,406,753,598,722,652,295,610,220,537,496,159,518,725,347,525,162,524,330,163,526,708,784,729,748,467,407,387,322,217,423,199,198,593,696,706,772,567,183,566,677,482,466,132,463,281,682,762,597,320,638,282,148,493,641,660,707,499,241,54,239,645,780,553,202,44,200,596,750,774,709,502,695,331,108,405,242,302,75)(26,121,227,510,234,557,769,441,655,246,632,559,335,123)(122,348,371,588,620,765,449,125,447,457,131,461,444,306)(155,299,606,408,656,661,508,778,523,771,704,633,723,475)(178,555,735,464,770,442,670,259,669,470,714,462,560,558)
comment:Define the group as a permutation group magma:G := PermutationGroup< 784 \| (1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440), (1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495), (1,2,8,38,173,301,692,590,238,89,308,577,754,758,726,513,683,309,99,261,60,12,58,252,602,203,601,362,711,717,474,327,421,386,101,21,4,18,87,346,727,742,619,298,321,80,319,492,147,491,317,79,16,3,13,63,151,32,149,495,365,93,363,584,500,611,206,45,204,315,113,139,478,581,775,529,266,546,540,713,579,545,172,37,170,538,548,174,456,600,768,571,221,91,358,738,721,702,384,100,184,176,552,284,685,434,271,316,554,251,180,39,49,193,42,191,487,477,585,703,712,318,326,614,465,776,570,185,40,182,458,129,379,153,438,289,550,511,691,700,303,424,763,777,522,674,265,672,751,693,767,668,257,115,255,392,195,314,646,760,615,208,294,285,311,400,625,628,361,697,426,313,643,678,533,451,730,352,433,118,25,5)(6,28,130,188,136,97,74,71,292,425,663,369,389,745,437,119,62,264,245,55,11,53,114,24,111,416,380,443,360,689,391,747,396,104,22,20,19,92,359,739,446,650,681,378,98,377,488,665,719,757,676,267,659,607,273,453,503,382,504,250,194,256,169,61,35,7,33,154,142,483,460,535,734,355,521,653,243,489,561,179,534,375,160,519,556,279,385,333,225,236,201,397,715,344,86,342,291,161,34,158,338,698,589,782,746,609,627,222,626,226,50,223,383,258,616,572,343,662,634,254,664,413,399,137,29,135,472,569,666,667,608,205,415,112,419,756,411,576,728,390,102,388,673,445,124,420,152,275,65,214,47,213,228,51,10,48,216,622,484,300,351,305,76,304,287,69,277,224,329,401,260,73,70,14,67,280,479,501,642,244,141,30)(9,43,196,587,404,694,490,146,31,144,428,209,117,430,233,52,231,418,190,68,283,543,171,356,90,354,732,733,417,480,744,455,127,27,126,450,373,544,297,272,64,270,435,595,473,568,759,737,675,679,394,631,710,724,341,541,647,240,219,367,720,699,741,575,494,189,41,187,563,701,617,211,507,248,56,181,107,403,687,286,686,761,755,409,109,23,106,215,599,364,422,486,368,95,247,657,374,594,603,530,515,157,514,779,578,671,263,562,640,564,520,574,253,166,134,186,357,736,469,644,237,506,636,432,573,498,150,497,290,637,229,635,340,459,429,116,353,731,604,512,156,509,439,766,591,269,531,296,528,164,527,621,337,84,17,82,328,345,128,427,764,618,705,307,454,629,262,623,752,690,288,688,783,592,197,381,743,649,586,336,210,46)(15,72,145,339,536,278,66,276,448,532,684,630,565,680,481,140,96,370,168,78,59,230,639,293,133,468,349,94,57,249,624,218,268,549,605,414,110,412,452,551,175,138,476,740,366,410,505,177,539,350,88,232,85,325,81,323,105,398,235,212,395,324,436,471,192,583,431,716,440,376,402,613,207,612,310,77,120,143,103,393,749,781,547,274,516,648,773,517,658,582,485,167,36,165,372,651,580,334,83,332,542,718,654,312,406,753,598,722,652,295,610,220,537,496,159,518,725,347,525,162,524,330,163,526,708,784,729,748,467,407,387,322,217,423,199,198,593,696,706,772,567,183,566,677,482,466,132,463,281,682,762,597,320,638,282,148,493,641,660,707,499,241,54,239,645,780,553,202,44,200,596,750,774,709,502,695,331,108,405,242,302,75)(26,121,227,510,234,557,769,441,655,246,632,559,335,123)(122,348,371,588,620,765,449,125,447,457,131,461,444,306)(155,299,606,408,656,661,508,778,523,771,704,633,723,475)(178,555,735,464,770,442,670,259,669,470,714,462,560,558) >; gap:G := Group( (1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440), (1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495), 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); sage:G = PermutationGroup(['(1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440)', 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sage_gap:G = gap.new('Group( 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)') oscar:G = @permutation_group(784, (1,3,14,68,58,253,261,231,640,526,265,62,126,451,772,657,736,712,664,481,144,488,713,363,616,681,596,217,104,22,4,19,93,359,739,780,748,391,102,21,99,380,427,718,419,760,631,248,660,685,429,503,153,502,720,337,540,647,529,164,375,321,702,734,764,491,485,142,30,139,479,567,189,244,168,36,7,34,159,66,277,117,431,401,477,287,342,725,700,530,397,571,522,160,520,193,543,180,562,418,268,63,222,498,170,539,404,732,717,689,753,515,576,426,115,24,112,393,138,29,135,185,569,257,667,608,310,110,413,757,570,251,662,563,684,378,204,439,454,716,758,595,275,619,274,621,699,697,296,625,219,70,289,692,304,701,303,75,300,201,294,279,350,328,225,295,72,216,71,293,334,653,779,641,561,601,355,136,468,147,31,6)(2,9,44,86,343,149,496,501,578,566,775,782,745,517,158,516,663,637,325,598,237,53,235,643,357,737,749,435,727,626,568,184,533,166,531,726,456,127,453,751,450,729,744,766,683,281,673,512,542,638,229,51,163,35,162,47,194,42,192,82,329,165,458,665,483,452,403,301,445,587,242,54,211,46,208,614,783,592,593,381,241,649,586,195,585,690,594,198,525,629,223,221,486,143,188,416,693,639,385,430,232,400,389,746,405,292,695,698,297,482,140,480,383,682,545,354,604,750,459,129,264,353,89,37,171,541,434,118,432,152,32,150,414,506,710,316,398,267,675,532,436,272,169,218,48,57,250,213,252,148,494,489,145,346,728,351,407,752,721,659,247,658,613,705,422,113,421,761,755,410,109,402,106,215,611,362,687,490,366,94,56,11)(5,23,107,207,45,205,606,392,146,105,399,636,781,620,214,574,624,552,564,181,449,535,167,534,345,91,60,259,280,583,365,591,719,730,558,754,572,186,40,183,260,442,361,738,514,519,518,715,588,196,527,548,331,528,437,765,784,731,650,762,617,386,508,396,318,199,43,197,590,246,55,243,652,671,282,356,735,597,200,509,733,467,132,464,269,677,575,276,642,607,441,768,743,623,239,646,446,771,500,603,463,747,455,774,461,504,283,330,577,190,262,306,76,15,73,41,8,39,178,556,493,668,724,390,538,670,284,111,90,18,88,179,560,546,173,209,67,133,28,131,95,367,352,648,240,609,444,605,759,415,212,618,465,475,137,474,740,368,409,384,227,50,224,370,233,471,134,470,395,103,394,469,412,323,714,341,85,17,83,333,124,26)(10,49,220,228,634,769,644,678,290,245,654,249,661,270,672,376,97,374,339,633,230,307,484,696,742,373,408,108,406,420,635,513,156,510,377,369,651,266,425,741,557,203,141,364,317,258,476,555,686,387,101,98,20,96,371,187,573,238,65,13,33,155,79,315,130,460,711,319,299,74,298,627,600,473,157,122,332,92,360,584,202,599,770,612,388,492,286,521,478,559,263,61,12,59,256,443,123,417,172,544,226,630,524,723,340,151,499,291,694,372,656,537,507,154,505,379,497,704,773,680,273,64,271,679,632,411,589,536,628,338,84,335,487,236,210,424,114,423,669,688,551,776,756,472,722,457,128,27,100,382,674,622,778,763,615,344,305,703,511,523,161,438,119,25,116,428,125,448,666,254,255,120,336,462,645,676,691,288,69,285,234,52)(16,77,308,358,433,322,550,777,553,176,38,174,175,80)(78,312,278,327,466,320,206,610,565,580,326,81,324,314)(87,347,547,708,582,767,447,182,349,709,549,302,495,348)(121,177,309,311,602,313,707,655,706,554,581,191,579,440), (1,4,20,97,375,544,292,680,284,68,37,7)(2,10,50,225,106,401,539,625,374,266,62,12)(3,15,74,192,530,165,517,430,564,345,86,17)(5,24,113,294,71,14,69,286,664,376,129,27)(6,29,136,473,382,502,691,755,615,504,153,32)(8,40,184,548,426,474,763,683,321,149,195,42)(9,45,83,152,199,595,583,612,594,621,215,47)(11,54,240,239,218,623,657,367,411,109,251,57)(13,64,147,130,434,298,556,277,343,515,279,66)(16,78,313,708,433,709,314,349,87,312,327,81)(18,89,352,713,318,79,316,289,693,611,252,91)(19,94,261,427,115,425,255,58,254,665,369,95)(21,100,383,141,396,219,48,217,601,732,379,98)(22,103,52,232,641,512,761,766,642,233,400,105)(23,108,407,498,629,587,626,617,577,402,415,110)(25,117,101,385,672,392,102,389,608,231,378,120)(26,122,441,125)(28,132,465,262,60,260,590,538,437,446,124,134)(30,140,237,235,133,469,357,90,355,146,487,143)(31,145,405,578,190,41,188,575,522,167,161,148)(33,156,511,476,634,524,497,722,339,84,336,157)(34,160,521,288,689,499,727,573,768,616,208,139)(35,138,477,736,742,756,570,238,53,236,137,164)(36,166,532,410,409,724,637,534,647,738,537,169)(38,175,550,324,176,302,602,278,581,553,554,177)(39,179,384,730,609,205,607,356,715,323,386,181)(43,198,370,701,731,688,663,253,614,729,351,88)(44,201,598,480,682,518,733,354,186,287,603,203)(46,209,418,111,417,545,481,746,639,690,412,212)(49,221,61,55,244,649,561,772,529,694,628,222)(51,72,171,542,593,197,591,297,73,296,173,230)(56,247,264,618,552,241,650,748,743,773,452,126)(59,257,210,585,666,546,717,329,380,488,328,258)(63,267,170,540,275,256,226,631,627,638,678,269)(65,274,492,592,315,214,619,751,397,104,342,116)(67,281,509,293,677,270,432,248,659,435,118,282)(70,290,158,406,754,520,533,216,571,185,472,291)(75,301,699,758,413,711,317,112,420,213,307,76)(77,309,706,358,322,80,320,308,582,191,580,311)(82,330,718,574,187,568,359,150,346,516,719,331)(85,340,127,454,445,459,456,471,519,398,394,159)(92,361,712,489,662,576,189,114,220,93,364,362)(96,372,741,599,428,622,679,319,423,443,249,373)(99,162,223,613,528,207,163,107,404,527,377,381)(119,436,172,341,265,673,451,697,503,228,245,439)(123,442,523,735,669,670,457,588,770,620,723,444)(128,353,667,450,458,695,390,648,494,624,684,283)(131,462,461,661,765,769,560,299,470,555,558,371)(135,200,263,566,431,675,783,710,333,671,775,463)(142,484,744,388,605,204,604,692,734,491,513,460)(144,479,334,674,635,700,344,726,438,280,653,416)(151,500,757,782,739,562,419,202,600,779,776,501)(154,506,676,784,760,737,702,304,303,271,305,300)(155,508,656,246,510,606,234,259,335,449,408,464)(168,536,658,387,395,725,482,660,310,268,654,505)(174,547,206,347,182,565,326,466,777,610,579,549)(178,557,306,704,714,778,475,633,227,632,771,559)(180,563,363,338,584,193,360,728,589,196,569,525)(183,368,366,652,764,759,686,698,543,421,414,483)(194,507,535,485,721,337,685,391,703,424,681,273)(211,393,750,541,272,455,399,752,668,567,243,448)(224,332,705,596,749,531,229,636,747,403,365,350)(242,551,597,468,325,716,780,526,630,696,295,651)(250,486,646,276,453,740,429,493,645,490,720,586)(422,514,640,572,644,643,753,745,496,687,467,762)(440,767,707,495), (1,2,8,38,173,301,692,590,238,89,308,577,754,758,726,513,683,309,99,261,60,12,58,252,602,203,601,362,711,717,474,327,421,386,101,21,4,18,87,346,727,742,619,298,321,80,319,492,147,491,317,79,16,3,13,63,151,32,149,495,365,93,363,584,500,611,206,45,204,315,113,139,478,581,775,529,266,546,540,713,579,545,172,37,170,538,548,174,456,600,768,571,221,91,358,738,721,702,384,100,184,176,552,284,685,434,271,316,554,251,180,39,49,193,42,191,487,477,585,703,712,318,326,614,465,776,570,185,40,182,458,129,379,153,438,289,550,511,691,700,303,424,763,777,522,674,265,672,751,693,767,668,257,115,255,392,195,314,646,760,615,208,294,285,311,400,625,628,361,697,426,313,643,678,533,451,730,352,433,118,25,5)(6,28,130,188,136,97,74,71,292,425,663,369,389,745,437,119,62,264,245,55,11,53,114,24,111,416,380,443,360,689,391,747,396,104,22,20,19,92,359,739,446,650,681,378,98,377,488,665,719,757,676,267,659,607,273,453,503,382,504,250,194,256,169,61,35,7,33,154,142,483,460,535,734,355,521,653,243,489,561,179,534,375,160,519,556,279,385,333,225,236,201,397,715,344,86,342,291,161,34,158,338,698,589,782,746,609,627,222,626,226,50,223,383,258,616,572,343,662,634,254,664,413,399,137,29,135,472,569,666,667,608,205,415,112,419,756,411,576,728,390,102,388,673,445,124,420,152,275,65,214,47,213,228,51,10,48,216,622,484,300,351,305,76,304,287,69,277,224,329,401,260,73,70,14,67,280,479,501,642,244,141,30)(9,43,196,587,404,694,490,146,31,144,428,209,117,430,233,52,231,418,190,68,283,543,171,356,90,354,732,733,417,480,744,455,127,27,126,450,373,544,297,272,64,270,435,595,473,568,759,737,675,679,394,631,710,724,341,541,647,240,219,367,720,699,741,575,494,189,41,187,563,701,617,211,507,248,56,181,107,403,687,286,686,761,755,409,109,23,106,215,599,364,422,486,368,95,247,657,374,594,603,530,515,157,514,779,578,671,263,562,640,564,520,574,253,166,134,186,357,736,469,644,237,506,636,432,573,498,150,497,290,637,229,635,340,459,429,116,353,731,604,512,156,509,439,766,591,269,531,296,528,164,527,621,337,84,17,82,328,345,128,427,764,618,705,307,454,629,262,623,752,690,288,688,783,592,197,381,743,649,586,336,210,46)(15,72,145,339,536,278,66,276,448,532,684,630,565,680,481,140,96,370,168,78,59,230,639,293,133,468,349,94,57,249,624,218,268,549,605,414,110,412,452,551,175,138,476,740,366,410,505,177,539,350,88,232,85,325,81,323,105,398,235,212,395,324,436,471,192,583,431,716,440,376,402,613,207,612,310,77,120,143,103,393,749,781,547,274,516,648,773,517,658,582,485,167,36,165,372,651,580,334,83,332,542,718,654,312,406,753,598,722,652,295,610,220,537,496,159,518,725,347,525,162,524,330,163,526,708,784,729,748,467,407,387,322,217,423,199,198,593,696,706,772,567,183,566,677,482,466,132,463,281,682,762,597,320,638,282,148,493,641,660,707,499,241,54,239,645,780,553,202,44,200,596,750,774,709,502,695,331,108,405,242,302,75)(26,121,227,510,234,557,769,441,655,246,632,559,335,123)(122,348,371,588,620,765,449,125,447,457,131,461,444,306)(155,299,606,408,656,661,508,778,523,771,704,633,723,475)(178,555,735,464,770,442,670,259,669,470,714,462,560,558))

Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SOPlus(4,13)$.

Homology

Abelianization:	$C_{2} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator()) sage_gap:G.FactorGroup(G.DerivedSubgroup()) oscar:quo(G, derived_subgroup(G)[1])
Schur multiplier:	not computed	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$1$	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

oscar:subgroups(G)

Subgroup data has not been computed.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table

Complex character table

See the $200 \times 200$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $86 \times 86$ rational character table.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	4769856.a
Order	\( 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \)
Exponent	\( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)

Nilpotent	no
Solvable	no
$\card{G^{\mathrm{ab}}}$	\( 2 \)
$\card{Z(G)}$	2
$\card{\Aut(G)}$	\( 2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \)
$\card{\mathrm{Out}(G)}$	\( 2^{3} \)
Perm deg.	$784$
Trans deg.	not computed
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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