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Properties

Label 821...000.a
Order \( 2^{19} \cdot 3^{4} \cdot 5^{3} \cdot 7 \cdot 13 \cdot 17 \)
Exponent \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{19} \cdot 3^{4} \cdot 5^{3} \cdot 7 \cdot 13 \cdot 17 \)
$\card{\mathrm{Out}(G)}$ \( 1 \)
Perm deg. $1365$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := PSigmaSp(6,4);
 
Copy content comment:Define the group as a permutation group
 
Copy content gap:G := Group( (1,102,176,740,103,73,658,769,874,981,1340,5,6,3,2)(4,1339,1309,1337,671,688,973,687,72,561,869,795,642,44,1154)(7,790,318,972,1025,866,379,1221,417,100,98,1096,1133,483,605)(8,1067,451,937,1273,1190,567,390,505,774,1102,1014,1248,1138,910)(9,477,1292,1145,1209,969,876,349,161,1091,1217,765,1086,1117,543)(10,1175,1289,523,761,911,962,1056,1165,328,571,1124,190,38,880)(11,1110,64,133,923,66,1008,119,849,1004,310,1193,585,901,154)(12,392,208,1061,372,40,699,648,181,309,434,997,694,124,1226)(13,1205,388,786,835,33,1365,385,500,639,1185,1026,433,537,606)(14,421,946,1250,269,728,327,1345,1095,143,58,211,420,334,1072)(15,666,217,636,185,164,439,321,1203,1040,1207,1045,729,1169,227)(16,166,715,257,676,759,1243,429,281,447,635,419,1213,993,896)(17,1263,985,69,847)(18,1224,1253,1088,411,918,401,1182,353,587,998,533,690,125,854)(19,680,207,1197,830,1109,821,797,240,1299,1298,542,1296,63,942)(20,262,1022,859,831,132,270,685,1163,913,343,1347,49,954,971)(21,94,296,707,491,249,1242,1230,792,292,183,1054,895,1210,311)(22,1234,224,766,663,1194,469,1268,519,340,828,721,395,719,96)(23,304,626,1137,1334,199,575,619,508,440,656,700,31,1330,546)(24,924,155,48,258,1058,532,595,213,91,762,479,1113,1223,853)(25,450,1071,902,399,733,216,99,801,1233,826,1363,773,404,1295)(26,882,369,842,1128,1159,815,887,530,1164,512,573,285,1172,126)(27,74,577,931,282,423,414,1105,490,779,1130,115,922,755,764)(28,283,569,223,456,1290,362,548,758,535,768,597,776,312,319)(29,871,714,272,123,572,727,128,493,947,957,904,1218,995,1362)(30,1341,1239,204,186,168,531,767,268,182,222,426,559,1010,350)(32,299,1039,60,824,725,870,738,1236,504,541,1189,627,323,400)(34,1254,1282,516,1235,243,932,756,832,945,1114,473,70,1141,1343)(35,80,965,906,864,811,536,1286,46,558,197,806,1241,1349,111)(36,784,1093,198,1306,397,1276,170,374,580,1219,237,916,802,1249)(37,169,1020,485,386,963,1291,274,273,1227,271,253,1046,325,624)(39,1348,351,529,148,1112,245,110,1310,1311,220,614,883,852,661)(41,120,603,341,808,256,739,1199,1121,843,950,226,1354,81,978)(42,466,641)(43,1356,787,202,1308,177,313,444,1355,104,673,771,670,669,640)(45,528,470,361,95,1316,524,476,1160,712,1007,539,1052,1270,914)(47,925,695,681,289,664,130,1157,235,1155,418,935,607,612,565)(50,286,620,1215,206,67,442,798,360,322,1127,616,218,851,1116)(51,137,136,1031,413,703,487,474,463,1070,355,654,722,622,1335)(52,436,1186,1048,499,255,930,335,634,939,643,203,748,592,79)(53,1162,514,109,195,809,138,1115,526,492,496,305,615,230,82)(54,1220,1202,149,646,1043,1358,1097,977,796,696,326,525,1034,934)(55,684,1261,717,251,772,1251,1050,1135,121,393,153,482,1006,649)(56,290,1142,979,75,1122,1275,848,192,1246,1151,1353,187,827,330)(57,61,191,608,406,578,424,380,837,598,789,1013,628,1083,1208)(59,885,741,117,1214,763,280,839,448,87,996,263,510,879,62)(65,454,1271,804,599,549,1229,938,527,576,368,810,384,288,683)(68,718,364,1041,522,1148,974,348,1364,581,76,1101,116,145,655)(71,478,150,704,84,232,291,547,1285,917,594,912,709,898,1032)(77,373,584,112,1132,250,371,716,1015,1260,890,1073,363,139,219)(78,660,1166,83,723,976,173,1098,518,781,570,178,484,359,1360)(85,846,431,833,921,461,1302,1134,653,1361,630,344,861,1019,596)(86,276,475,583,336,1180,151,1191,645,347,494,215,1018,805,1051)(88,1012,142,409,377,275,697,713,657,1016,967,193,498,1158,300)(89,820,108,1090,383,1288,980,1136,855,443,1168,662,961,1222,550)(90,1064,566,836,194,1075,165,1036,1055,625,926,234,134,1312,1187)(92,520,1237,943,878,1103,1074,1011,441,905,850,920,749,267,1108)(93,229,513,179,1359)(97,307,557,412,760,940,990,1044,105,521,1204,750,560,959,889)(101,873,672,875,314)(106,867,698,375,345,1085,452,1262,693,724,1021,1100,1225,1106,320)(107,788,823,506,1212,590,814,677,1174,692,302,1265,829,731,1318)(113,735,1000,730,1065,295,780,1294,933,745,1047,156,114,564,1077)(118,1192,752,389,435,1200,1297,503,1035,1315,205,428,1053,617,734)(122,1171,952,147,919,538,1238,1267,894,480,471,822,297,1030,1028)(127,720,398,1300,984,897,858,376,140,1277,1332,1331,602,674,989)(129,1084,1188,799,254,331,1099,236,838,306,955,817,708,515,509)(131,1059,159,184,287,1350,553,1123,1033,563,1027,1150,1252,891,845)(135,1305,747,1240,1023,545,678,785,604,458,1206,261,1176,701,301)(141,964,819,944,1231,679,1140,744,1125,966,659,333,346,316,511)(144,1139,1198,405,1272)(152,467,983,501,746,242,588,1328,593,652,317,884,552,986,791)(157,252,647,865,705,460,495,244,214,1009,278,298,737,1152,586)(158,1082,941,994,751,338,1120,1279,354,507,1147,582,951,1003,907)(160,1352,464,279,736,1119,1327,960,1259,621,591,366,928,407,857)(162,1344,453,488,481)(163,893,800,556,172,1320,438,988,1201,1177,812,958,982,1062,472)(167,1324,825,1005,1351,1266,265,840,1017,903,358,629,246,1060,783)(171,1118,239,872,909,644,794,1281,1303,610,1104,1042,266,1049,1146)(174,324,238,579,1068,1287,1357,631,284,1244,860,1216,574,757,459)(175,1170,600,1111,665,1326,668,908,1066,1278,264,303,465,209,534)(180,929,743,1329,352)(188,1080,367,1024,332,370,1314,432,742,1144,618,1323,337,1076,1319)(189,540,468,308,1301,233,127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);
 
Copy content sage:G = PermutationGroup(['(1,102,176,740,103,73,658,769,874,981,1340,5,6,3,2)(4,1339,1309,1337,671,688,973,687,72,561,869,795,642,44,1154)(7,790,318,972,1025,866,379,1221,417,100,98,1096,1133,483,605)(8,1067,451,937,1273,1190,567,390,505,774,1102,1014,1248,1138,910)(9,477,1292,1145,1209,969,876,349,161,1091,1217,765,1086,1117,543)(10,1175,1289,523,761,911,962,1056,1165,328,571,1124,190,38,880)(11,1110,64,133,923,66,1008,119,849,1004,310,1193,585,901,154)(12,392,208,1061,372,40,699,648,181,309,434,997,694,124,1226)(13,1205,388,786,835,33,1365,385,500,639,1185,1026,433,537,606)(14,421,946,1250,269,728,327,1345,1095,143,58,211,420,334,1072)(15,666,217,636,185,164,439,321,1203,1040,1207,1045,729,1169,227)(16,166,715,257,676,759,1243,429,281,447,635,419,1213,993,896)(17,1263,985,69,847)(18,1224,1253,1088,411,918,401,1182,353,587,998,533,690,125,854)(19,680,207,1197,830,1109,821,797,240,1299,1298,542,1296,63,942)(20,262,1022,859,831,132,270,685,1163,913,343,1347,49,954,971)(21,94,296,707,491,249,1242,1230,792,292,183,1054,895,1210,311)(22,1234,224,766,663,1194,469,1268,519,340,828,721,395,719,96)(23,304,626,1137,1334,199,575,619,508,440,656,700,31,1330,546)(24,924,155,48,258,1058,532,595,213,91,762,479,1113,1223,853)(25,450,1071,902,399,733,216,99,801,1233,826,1363,773,404,1295)(26,882,369,842,1128,1159,815,887,530,1164,512,573,285,1172,126)(27,74,577,931,282,423,414,1105,490,779,1130,115,922,755,764)(28,283,569,223,456,1290,362,548,758,535,768,597,776,312,319)(29,871,714,272,123,572,727,128,493,947,957,904,1218,995,1362)(30,1341,1239,204,186,168,531,767,268,182,222,426,559,1010,350)(32,299,1039,60,824,725,870,738,1236,504,541,1189,627,323,400)(34,1254,1282,516,1235,243,932,756,832,945,1114,473,70,1141,1343)(35,80,965,906,864,811,536,1286,46,558,197,806,1241,1349,111)(36,784,1093,198,1306,397,1276,170,374,580,1219,237,916,802,1249)(37,169,1020,485,386,963,1291,274,273,1227,271,253,1046,325,624)(39,1348,351,529,148,1112,245,110,1310,1311,220,614,883,852,661)(41,120,603,341,808,256,739,1199,1121,843,950,226,1354,81,978)(42,466,641)(43,1356,787,202,1308,177,313,444,1355,104,673,771,670,669,640)(45,528,470,361,95,1316,524,476,1160,712,1007,539,1052,1270,914)(47,925,695,681,289,664,130,1157,235,1155,418,935,607,612,565)(50,286,620,1215,206,67,442,798,360,322,1127,616,218,851,1116)(51,137,136,1031,413,703,487,474,463,1070,355,654,722,622,1335)(52,436,1186,1048,499,255,930,335,634,939,643,203,748,592,79)(53,1162,514,109,195,809,138,1115,526,492,496,305,615,230,82)(54,1220,1202,149,646,1043,1358,1097,977,796,696,326,525,1034,934)(55,684,1261,717,251,772,1251,1050,1135,121,393,153,482,1006,649)(56,290,1142,979,75,1122,1275,848,192,1246,1151,1353,187,827,330)(57,61,191,608,406,578,424,380,837,598,789,1013,628,1083,1208)(59,885,741,117,1214,763,280,839,448,87,996,263,510,879,62)(65,454,1271,804,599,549,1229,938,527,576,368,810,384,288,683)(68,718,364,1041,522,1148,974,348,1364,581,76,1101,116,145,655)(71,478,150,704,84,232,291,547,1285,917,594,912,709,898,1032)(77,373,584,112,1132,250,371,716,1015,1260,890,1073,363,139,219)(78,660,1166,83,723,976,173,1098,518,781,570,178,484,359,1360)(85,846,431,833,921,461,1302,1134,653,1361,630,344,861,1019,596)(86,276,475,583,336,1180,151,1191,645,347,494,215,1018,805,1051)(88,1012,142,409,377,275,697,713,657,1016,967,193,498,1158,300)(89,820,108,1090,383,1288,980,1136,855,443,1168,662,961,1222,550)(90,1064,566,836,194,1075,165,1036,1055,625,926,234,134,1312,1187)(92,520,1237,943,878,1103,1074,1011,441,905,850,920,749,267,1108)(93,229,513,179,1359)(97,307,557,412,760,940,990,1044,105,521,1204,750,560,959,889)(101,873,672,875,314)(106,867,698,375,345,1085,452,1262,693,724,1021,1100,1225,1106,320)(107,788,823,506,1212,590,814,677,1174,692,302,1265,829,731,1318)(113,735,1000,730,1065,295,780,1294,933,745,1047,156,114,564,1077)(118,1192,752,389,435,1200,1297,503,1035,1315,205,428,1053,617,734)(122,1171,952,147,919,538,1238,1267,894,480,471,822,297,1030,1028)(127,720,398,1300,984,897,858,376,140,1277,1332,1331,602,674,989)(129,1084,1188,799,254,331,1099,236,838,306,955,817,708,515,509)(131,1059,159,184,287,1350,553,1123,1033,563,1027,1150,1252,891,845)(135,1305,747,1240,1023,545,678,785,604,458,1206,261,1176,701,301)(141,964,819,944,1231,679,1140,744,1125,966,659,333,346,316,511)(144,1139,1198,405,1272)(152,467,983,501,746,242,588,1328,593,652,317,884,552,986,791)(157,252,647,865,705,460,495,244,214,1009,278,298,737,1152,586)(158,1082,941,994,751,338,1120,1279,354,507,1147,582,951,1003,907)(160,1352,464,279,736,1119,1327,960,1259,621,591,366,928,407,857)(162,1344,453,488,481)(163,893,800,556,172,1320,438,988,1201,1177,812,958,982,1062,472)(167,1324,825,1005,1351,1266,265,840,1017,903,358,629,246,1060,783)(171,1118,239,872,909,644,794,1281,1303,610,1104,1042,266,1049,1146)(174,324,238,579,1068,1287,1357,631,284,1244,860,1216,574,757,459)(175,1170,600,1111,665,1326,668,908,1066,1278,264,303,465,209,534)(180,929,743,1329,352)(188,1080,367,1024,332,370,1314,432,742,1144,618,1323,337,1076,1319)(189,540,4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Group information

Description:$\PSigmaSp(6,4)$
Order: \(8212119552000\)\(\medspace = 2^{19} \cdot 3^{4} \cdot 5^{3} \cdot 7 \cdot 13 \cdot 17 \)
Copy content comment:Order of the group
 
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Exponent: \(1113840\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
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Automorphism group:Group of order \(8212119552000\)\(\medspace = 2^{19} \cdot 3^{4} \cdot 5^{3} \cdot 7 \cdot 13 \cdot 17 \)
Copy content comment:Automorphism group
 
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Composition factors:$C_2$, $\Sp(6,4)$
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Derived length:$1$
Copy content comment:Derived length of the group
 
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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);
 
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Copy content comment:Determine if the group G is cyclic
 
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Copy content comment:Determine if the group G is nilpotent
 
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Copy content comment:Determine if the group G is solvable
 
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Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
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Copy content comment:Determine if the group G is simple
 
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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 16 17 18 20 21 24 30 34 51 63 65 85
Elements 1 20654335 404218880 18979833600 8696346624 156393036800 65175552000 278014464000 65175552000 388792535040 940972032000 126340300800 586579968000 492727173120 513257472000 16102195200 456228864000 615908966400 130351104000 684343296000 821211955200 241532928000 322043904000 391053312000 505361203200 386452684800 8212119552000
Conjugacy classes   1 5 3 9 5 10 1 7 1 10 10 1 1 8 2 2 1 4 1 3 5 2 2 3 4 4 105
Divisions 1 5 3 9 5 10 1 6 1 10 10 1 1 8 1 1 1 4 1 2 5 1 1 1 1 1 91
Autjugacy classes 1 5 3 9 5 10 1 7 1 10 10 1 1 8 2 2 1 4 1 3 5 2 2 3 4 4 105

Minimal presentations

Permutation degree:$1365$
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 378 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\PSigmaSp(6,4)$
Copy content magma:G := PSigmaSp(6,4);
 
Copy content gap:G := Group([[[0*Z(4), Z(4)^0, Z(4)^0, Z(4)^0, 0*Z(4), 0*Z(4)], [Z(4)^0, 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4)], [0*Z(4), Z(4)^0, 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4)], [0*Z(4), 0*Z(4), Z(4)^0, 0*Z(4), Z(4)^0, 0*Z(4)], [0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), Z(4)^0], [0*Z(4), 0*Z(4), Z(4)^0, 0*Z(4), 0*Z(4), 0*Z(4)]],[[Z(4)^1, 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4)], [0*Z(4), Z(4)^0, 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4)], [0*Z(4), 0*Z(4), Z(4)^1, 0*Z(4), 0*Z(4), 0*Z(4)], [0*Z(4), 0*Z(4), 0*Z(4), Z(4)^2, 0*Z(4), 0*Z(4)], [0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), Z(4)^0, 0*Z(4)], [0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), 0*Z(4), Z(4)^2]]]);
 
Copy content sage:F = GF(4); al = F.0; MS = MatrixSpace(F, 6, 6) G = MatrixGroup([MS([[0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0]]), MS([[al^1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, al^1, 0, 0, 0], [0, 0, 0, al^2, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, al^2]])])
 
Permutation group:Degree $1365$ $\langle(1,102,176,740,103,73,658,769,874,981,1340,5,6,3,2)(4,1339,1309,1337,671,688,973,687,72,561,869,795,642,44,1154) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 1365 | (1,102,176,740,103,73,658,769,874,981,1340,5,6,3,2)(4,1339,1309,1337,671,688,973,687,72,561,869,795,642,44,1154)(7,790,318,972,1025,866,379,1221,417,100,98,1096,1133,483,605)(8,1067,451,937,1273,1190,567,390,505,774,1102,1014,1248,1138,910)(9,477,1292,1145,1209,969,876,349,161,1091,1217,765,1086,1117,543)(10,1175,1289,523,761,911,962,1056,1165,328,571,1124,190,38,880)(11,1110,64,133,923,66,1008,119,849,1004,310,1193,585,901,154)(12,392,208,1061,372,40,699,648,181,309,434,997,694,124,1226)(13,1205,388,786,835,33,1365,385,500,639,1185,1026,433,537,606)(14,421,946,1250,269,728,327,1345,1095,143,58,211,420,334,1072)(15,666,217,636,185,164,439,321,1203,1040,1207,1045,729,1169,227)(16,166,715,257,676,759,1243,429,281,447,635,419,1213,993,896)(17,1263,985,69,847)(18,1224,1253,1088,411,918,401,1182,353,587,998,533,690,125,854)(19,680,207,1197,830,1109,821,797,240,1299,1298,542,1296,63,942)(20,262,1022,859,831,132,270,685,1163,913,343,1347,49,954,971)(21,94,296,707,491,249,1242,1230,792,292,183,1054,895,1210,311)(22,1234,224,766,663,1194,469,1268,519,340,828,721,395,719,96)(23,304,626,1137,1334,199,575,619,508,440,656,700,31,1330,546)(24,924,155,48,258,1058,532,595,213,91,762,479,1113,1223,853)(25,450,1071,902,399,733,216,99,801,1233,826,1363,773,404,1295)(26,882,369,842,1128,1159,815,887,530,1164,512,573,285,1172,126)(27,74,577,931,282,423,414,1105,490,779,1130,115,922,755,764)(28,283,569,223,456,1290,362,548,758,535,768,597,776,312,319)(29,871,714,272,123,572,727,128,493,947,957,904,1218,995,1362)(30,1341,1239,204,186,168,531,767,268,182,222,426,559,1010,350)(32,299,1039,60,824,725,870,738,1236,504,541,1189,627,323,400)(34,1254,1282,516,1235,243,932,756,832,945,1114,473,70,1141,1343)(35,80,965,906,864,811,536,1286,46,558,197,806,1241,1349,111)(36,784,1093,198,1306,397,1276,170,374,580,1219,237,916,802,1249)(37,169,1020,485,386,963,1291,274,273,1227,271,253,1046,325,624)(39,1348,351,529,148,1112,245,110,1310,1311,220,614,883,852,661)(41,120,603,341,808,256,739,1199,1121,843,950,226,1354,81,978)(42,466,641)(43,1356,787,202,1308,177,313,444,1355,104,673,771,670,669,640)(45,528,470,361,95,1316,524,476,1160,712,1007,539,1052,1270,914)(47,925,695,681,289,664,130,1157,235,1155,418,935,607,612,565)(50,286,620,1215,206,67,442,798,360,322,1127,616,218,851,1116)(51,137,136,1031,413,703,487,474,463,1070,355,654,722,622,1335)(52,436,1186,1048,499,255,930,335,634,939,643,203,748,592,79)(53,1162,514,109,195,809,138,1115,526,492,496,305,615,230,82)(54,1220,1202,149,646,1043,1358,1097,977,796,696,326,525,1034,934)(55,684,1261,717,251,772,1251,1050,1135,121,393,153,482,1006,649)(56,290,1142,979,75,1122,1275,848,192,1246,1151,1353,187,827,330)(57,61,191,608,406,578,424,380,837,598,789,1013,628,1083,1208)(59,885,741,117,1214,763,280,839,448,87,996,263,510,879,62)(65,454,1271,804,599,549,1229,938,527,576,368,810,384,288,683)(68,718,364,1041,522,1148,974,348,1364,581,76,1101,116,145,655)(71,478,150,704,84,232,291,547,1285,917,594,912,709,898,1032)(77,373,584,112,1132,250,371,716,1015,1260,890,1073,363,139,219)(78,660,1166,83,723,976,173,1098,518,781,570,178,484,359,1360)(85,846,431,833,921,461,1302,1134,653,1361,630,344,861,1019,596)(86,276,475,583,336,1180,151,1191,645,347,494,215,1018,805,1051)(88,1012,142,409,377,275,697,713,657,1016,967,193,498,1158,300)(89,820,108,1090,383,1288,980,1136,855,443,1168,662,961,1222,550)(90,1064,566,836,194,1075,165,1036,1055,625,926,234,134,1312,1187)(92,520,1237,943,878,1103,1074,1011,441,905,850,920,749,267,1108)(93,229,513,179,1359)(97,307,557,412,760,940,990,1044,105,521,1204,750,560,959,889)(101,873,672,875,314)(106,867,698,375,345,1085,452,1262,693,724,1021,1100,1225,1106,320)(107,788,823,506,1212,590,814,677,1174,692,302,1265,829,731,1318)(113,735,1000,730,1065,295,780,1294,933,745,1047,156,114,564,1077)(118,1192,752,389,435,1200,1297,503,1035,1315,205,428,1053,617,734)(122,1171,952,147,919,538,1238,1267,894,480,471,822,297,1030,1028)(127,720,398,1300,984,897,858,376,140,1277,1332,1331,602,674,989)(129,1084,1188,799,254,331,1099,236,838,306,955,817,708,515,509)(131,1059,159,184,287,1350,553,1123,1033,563,1027,1150,1252,891,845)(135,1305,747,1240,1023,545,678,785,604,458,1206,261,1176,701,301)(141,964,819,944,1231,679,1140,744,1125,966,659,333,346,316,511)(144,1139,1198,405,1272)(152,467,983,501,746,242,588,1328,593,652,317,884,552,986,791)(157,252,647,865,705,460,495,244,214,1009,278,298,737,1152,586)(158,1082,941,994,751,338,1120,1279,354,507,1147,582,951,1003,907)(160,1352,464,279,736,1119,1327,960,1259,621,591,366,928,407,857)(162,1344,453,488,481)(163,893,800,556,172,1320,438,988,1201,1177,812,958,982,1062,472)(167,1324,825,1005,1351,1266,265,840,1017,903,358,629,246,1060,783)(171,1118,239,872,909,644,794,1281,1303,610,1104,1042,266,1049,1146)(174,324,238,579,1068,1287,1357,631,284,1244,860,1216,574,757,459)(175,1170,600,1111,665,1326,668,908,1066,1278,264,303,465,209,534)(180,929,743,1329,352)(188,1080,367,1024,332,370,1314,432,742,1144,618,1323,337,1076,1319)(189,540,468,308,1301,233,1274,437,357,555,877,777,1092,782,818)(196,975,489,1342,956)(200,1057,775,710,1037,430,1228,863,886,868,1264,1063,1325,445,1161)(201,689,770,294,1338)(210,754,231,315,1087,502,856,544,968,816,1181,449,888,601,277)(212,915,248,382,247)(221,228,1245,1094,948,381,1069,241,651,589,446,1079,486,1211,1002)(225,562,403,455,1131,1256,497,1247,1029,632,259,637,1195,1089,1149)(260,1284,457,633,609,702,1129,1184,841,834,402,416,807,517,813)(293,1346,1232,686,422,682,1322,554,793,1307,691,650,1336,638,1258)(329,927,667,899,675,551,949,1107,1257,391,732,339,1321,387,342)(356,970,953,987,408)(365,726,753,862,706)(378,1333,396,881,427,1183,900,1293,1283,410,394,1038,778,1304,613)(415,936,1001,1173,1179)(425,803,568,1078,1156)(462,1081,1126,1255,1178)(611,711,991,844,992)(623,1269,1280,999,1196)(892,1317,1313,1143,1167), 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>;
 
Copy content gap:G := Group( 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);
 
Copy content sage:G = PermutationGroup(['(1,102,176,740,103,73,658,769,874,981,1340,5,6,3,2)(4,1339,1309,1337,671,688,973,687,72,561,869,795,642,44,1154)(7,790,318,972,1025,866,379,1221,417,100,98,1096,1133,483,605)(8,1067,451,937,1273,1190,567,390,505,774,1102,1014,1248,1138,910)(9,477,1292,1145,1209,969,876,349,161,1091,1217,765,1086,1117,543)(10,1175,1289,523,761,911,962,1056,1165,328,571,1124,190,38,880)(11,1110,64,133,923,66,1008,119,849,1004,310,1193,585,901,154)(12,392,208,1061,372,40,699,648,181,309,434,997,694,124,1226)(13,1205,388,786,835,33,1365,385,500,639,1185,1026,433,537,606)(14,421,946,1250,269,728,327,1345,1095,143,58,211,420,334,1072)(15,666,217,636,185,164,439,321,1203,1040,1207,1045,729,1169,227)(16,166,715,257,676,759,1243,429,281,447,635,419,1213,993,896)(17,1263,985,69,847)(18,1224,1253,1088,411,918,401,1182,353,587,998,533,690,125,854)(19,680,207,1197,830,1109,821,797,240,1299,1298,542,1296,63,942)(20,262,1022,859,831,132,270,685,1163,913,343,1347,49,954,971)(21,94,296,707,491,249,1242,1230,792,292,183,1054,895,1210,311)(22,1234,224,766,663,1194,469,1268,519,340,828,721,395,719,96)(23,304,626,1137,1334,199,575,619,508,440,656,700,31,1330,546)(24,924,155,48,258,1058,532,595,213,91,762,479,1113,1223,853)(25,450,1071,902,399,733,216,99,801,1233,826,1363,773,404,1295)(26,882,369,842,1128,1159,815,887,530,1164,512,573,285,1172,126)(27,74,577,931,282,423,414,1105,490,779,1130,115,922,755,764)(28,283,569,223,456,1290,362,548,758,535,768,597,776,312,319)(29,871,714,272,123,572,727,128,493,947,957,904,1218,995,1362)(30,1341,1239,204,186,168,531,767,268,182,222,426,559,1010,350)(32,299,1039,60,824,725,870,738,1236,504,541,1189,627,323,400)(34,1254,1282,516,1235,243,932,756,832,945,1114,473,70,1141,1343)(35,80,965,906,864,811,536,1286,46,558,197,806,1241,1349,111)(36,784,1093,198,1306,397,1276,170,374,580,1219,237,916,802,1249)(37,169,1020,485,386,963,1291,274,273,1227,271,253,1046,325,624)(39,1348,351,529,148,1112,245,110,1310,1311,220,614,883,852,661)(41,120,603,341,808,256,739,1199,1121,843,950,226,1354,81,978)(42,466,641)(43,1356,787,202,1308,177,313,444,1355,104,673,771,670,669,640)(45,528,470,361,95,1316,524,476,1160,712,1007,539,1052,1270,914)(47,925,695,681,289,664,130,1157,235,1155,418,935,607,612,565)(50,286,620,1215,206,67,442,798,360,322,1127,616,218,851,1116)(51,137,136,1031,413,703,487,474,463,1070,355,654,722,622,1335)(52,436,1186,1048,499,255,930,335,634,939,643,203,748,592,79)(53,1162,514,109,195,809,138,1115,526,492,496,305,615,230,82)(54,1220,1202,149,646,1043,1358,1097,977,796,696,326,525,1034,934)(55,684,1261,717,251,772,1251,1050,1135,121,393,153,482,1006,649)(56,290,1142,979,75,1122,1275,848,192,1246,1151,1353,187,827,330)(57,61,191,608,406,578,424,380,837,598,789,1013,628,1083,1208)(59,885,741,117,1214,763,280,839,448,87,996,263,510,879,62)(65,454,1271,804,599,549,1229,938,527,576,368,810,384,288,683)(68,718,364,1041,522,1148,974,348,1364,581,76,1101,116,145,655)(71,478,150,704,84,232,291,547,1285,917,594,912,709,898,1032)(77,373,584,112,1132,250,371,716,1015,1260,890,1073,363,139,219)(78,660,1166,83,723,976,173,1098,518,781,570,178,484,359,1360)(85,846,431,833,921,461,1302,1134,653,1361,630,344,861,1019,596)(86,276,475,583,336,1180,151,1191,645,347,494,215,1018,805,1051)(88,1012,142,409,377,275,697,713,657,1016,967,193,498,1158,300)(89,820,108,1090,383,1288,980,1136,855,443,1168,662,961,1222,550)(90,1064,566,836,194,1075,165,1036,1055,625,926,234,134,1312,1187)(92,520,1237,943,878,1103,1074,1011,441,905,850,920,749,267,1108)(93,229,513,179,1359)(97,307,557,412,760,940,990,1044,105,521,1204,750,560,959,889)(101,873,672,875,314)(106,867,698,375,345,1085,452,1262,693,724,1021,1100,1225,1106,320)(107,788,823,506,1212,590,814,677,1174,692,302,1265,829,731,1318)(113,735,1000,730,1065,295,780,1294,933,745,1047,156,114,564,1077)(118,1192,752,389,435,1200,1297,503,1035,1315,205,428,1053,617,734)(122,1171,952,147,919,538,1238,1267,894,480,471,822,297,1030,1028)(127,720,398,1300,984,897,858,376,140,1277,1332,1331,602,674,989)(129,1084,1188,799,254,331,1099,236,838,306,955,817,708,515,509)(131,1059,159,184,287,1350,553,1123,1033,563,1027,1150,1252,891,845)(135,1305,747,1240,1023,545,678,785,604,458,1206,261,1176,701,301)(141,964,819,944,1231,679,1140,744,1125,966,659,333,346,316,511)(144,1139,1198,405,1272)(152,467,983,501,746,242,588,1328,593,652,317,884,552,986,791)(157,252,647,865,705,460,495,244,214,1009,278,298,737,1152,586)(158,1082,941,994,751,338,1120,1279,354,507,1147,582,951,1003,907)(160,1352,464,279,736,1119,1327,960,1259,621,591,366,928,407,857)(162,1344,453,488,481)(163,893,800,556,172,1320,438,988,1201,1177,812,958,982,1062,472)(167,1324,825,1005,1351,1266,265,840,1017,903,358,629,246,1060,783)(171,1118,239,872,909,644,794,1281,1303,610,1104,1042,266,1049,1146)(174,324,238,579,1068,1287,1357,631,284,1244,860,1216,574,757,459)(175,1170,600,1111,665,1326,668,908,1066,1278,264,303,465,209,534)(180,929,743,1329,352)(188,1080,367,1024,332,370,1314,432,742,1144,618,1323,337,1076,1319)(189,540,4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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 equivalence classes (represented by square brackets) of matrices in $\SigmaSp(6,4)$.

Homology

Abelianization: $C_{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 $105 \times 105$ character table is not available for this group.

Rational character table

The $91 \times 91$ rational character table is not available for this group.