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Properties

Label 58968000000.c
Order \( 2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
Exponent \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
$\card{\mathrm{Out}(G)}$ \( 1 \)
Perm deg. $756$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

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

Group information

Description:$\PGammaU(4,5)$
Order: \(58968000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Exponent: \(32760\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Automorphism group:$\PGammaU(4,5)$, of order \(58968000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 2, $\PSU(4,5)$
Copy content comment:Composition factors of the group
 
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Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Derived length:$1$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

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

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
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Copy content sage_gap:G.IsCyclic()
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
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Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content 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;
 
Copy content 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");
 
Copy content 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))
 

Order 1 2 3 4 5 6 7 8 9 10 12 13 14 15 18 20 21 24 26 30 40 42 52 60 63 104 120 126
Elements 1 2391775 7192250 379260000 244140624 1703204750 468000000 3641400000 468000000 2915967600 5180175000 850500000 468000000 782964000 468000000 3243240000 936000000 8435700000 7654500000 5041764000 1965600000 936000000 1701000000 1474200000 2808000000 3402000000 982800000 2808000000 58968000000
Conjugacy classes   1 6 3 7 4 16 1 8 1 12 13 3 1 5 1 5 2 15 9 12 2 2 3 2 6 6 2 6 154
Divisions 1 6 3 7 4 16 1 8 1 12 13 1 1 5 1 5 1 10 3 12 2 1 1 2 1 1 1 1 121
Autjugacy classes 1 6 3 7 4 16 1 8 1 12 13 3 1 5 1 5 2 15 9 12 2 2 3 2 6 6 2 6 154

Minimal presentations

Permutation degree:$756$
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 105 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


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

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\GammaU(4,5)$.

Homology

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

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 

Subgroup data has not been computed.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

The $154 \times 154$ character table is not available for this group.

Rational character table

The $121 \times 121$ rational character table is not available for this group.