Finite Group DatabasebetaFinite Group Databasebeta| order | 36930 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊(D1231⋊C3) |
| unambiguous semidirect product | (C1231⋊C3)⋊D5 |
| unambiguous semidirect product | D6155⋊C3 |
| Fitting, socle, derived, Hall | C5×C1231 ⋊ C2×C3 |
| Hall | C5×C1231⋊C3 ⋊ C2 |
| Hall | D6155 ⋊ C3 |
| Hall | C1231⋊C3 ⋊ D5 |
| Sylow | C1231 ⋊ C3×D5 |
| Sylow | C5 ⋊ D1231⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C1231 | C2×C3 |
| LCS | G ▷ C5×C1231 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1236 |
| transitive degree | μt(G) = 6155 |
A minimal faithful permutation representation — 2 generators on 1236 points:
(2,3)(4,5)(7,11,12,8,9,13)(10,19,20,14,15,29)(16,32,33,22,23,46)(17,31,27,25,21,18)(24,51,52,35,36,69)(26,49,50,38,34,43)(28,30,48,40,41,44)(37,74,75,54,55,98)(39,72,73,57,53,66)(42,47,71,59,60,67)(45,87,85,63,64,62)(56,103,104,77,78,133)(58,101,102,80,76,95)(61,70,100,82,83,96)(65,116,117,88,89,91)(68,122,114,92,93,86)(79,138,139,106,107,174)(81,136,137,109,105,130)(84,99,135,111,112,131)(90,151,152,118,119,126)(94,158,156,123,124,121)(97,163,149,127,128,115)(108,179,180,141,142,221)(110,177,178,144,140,171)(113,134,176,146,147,172)(120,192,193,153,154,167)(125,199,200,159,160,162)(129,205,197,164,165,157)(132,210,190,168,169,150)(143,226,227,182,183,274)(145,224,225,185,181,218)(148,175,223,187,188,219)(155,239,240,194,195,214)(161,246,247,201,202,209)(166,253,251,206,207,204)(170,258,244,211,212,198)(173,263,237,215,216,191)(184,279,280,229,230,333)(186,277,278,232,228,271)(189,222,276,234,235,272)(196,292,293,241,242,267)(203,299,300,248,249,262)(208,306,307,254,255,257)(213,312,304,259,260,252)(217,317,297,264,265,245)(220,322,290,268,269,238)(231,338,339,282,283,398)(233,336,337,285,281,330)(236,275,335,287,288,331)(243,351,352,294,295,326)(250,358,359,301,302,321)(256,365,366,308,309,316)(261,372,370,313,314,311)(266,377,363,318,319,305)(270,382,356,323,324,298)(273,387,349,327,328,291)(284,403,404,341,342,469)(286,401,402,344,340,395)(289,334,400,346,347,396)(296,416,417,353,354,391)(303,423,424,360,361,386)(310,430,431,367,368,381)(315,437,438,373,374,376)(320,443,435,378,379,371)(325,448,428,383,384,364)(329,453,421,388,389,357)(332,458,414,392,393,350)(343,474,475,406,407,546)(345,472,473,409,405,466)(348,399,471,411,412,467)(355,487,488,418,419,462)(362,494,495,425,426,457)(369,501,502,432,433,452)(375,508,509,439,440,447)(380,515,513,444,445,442)(385,520,506,449,450,436)(390,525,499,454,455,429)(394,530,492,459,460,422)(397,535,485,463,464,415)(408,551,552,477,478,629)(410,549,550,480,476,543)(413,470,548,482,483,544)(420,564,565,489,490,539)(427,571,572,496,497,534)(434,578,579,503,504,529)(441,585,586,510,511,524)(446,592,593,516,517,519)(451,598,590,521,522,514)(456,603,583,526,527,507)(461,608,576,531,532,500)(465,613,569,536,537,493)(468,618,562,540,541,486)(479,634,635,554,555,718)(481,632,633,557,553,626)(484,547,631,559,560,627)(491,647,648,566,567,622)(498,654,655,573,574,617)(505,661,662,580,581,612)(512,668,669,587,588,607)(518,675,676,594,595,602)(523,682,680,599,600,597)(528,687,673,604,605,591)(533,692,666,609,610,584)(538,697,659,614,615,577)(542,702,652,619,620,570)(545,707,645,623,624,563)(556,723,724,637,638,813)(558,721,722,640,636,715)(561,630,720,642,643,716)(568,736,737,649,650,711)(575,743,744,656,657,706)(582,750,751,663,664,701)(589,757,758,670,671,696)(596,764,765,677,678,691)(601,771,772,683,684,686)(606,777,769,688,689,681)(611,782,762,693,694,674)(616,787,755,698,699,667)(621,792,748,703,704,660)(625,797,741,708,709,653)(628,802,734,712,713,646)(639,818,819,726,727,914)(641,816,817,729,725,810)(644,719,815,731,732,811)(651,831,832,738,739,806)(658,838,839,745,746,801)(665,845,846,752,753,796)(672,852,853,759,760,791)(679,859,860,766,767,786)(685,866,867,773,774,781)(690,873,871,778,779,776)(695,878,864,783,784,770)(700,883,857,788,789,763)(705,888,850,793,794,756)(710,893,843,798,799,749)(714,898,836,803,804,742)(717,903,829,807,808,735)(728,919,920,821,822,1021)(730,917,918,824,820,911)(733,814,916,826,827,912)(740,932,933,833,834,907)(747,939,940,840,841,902)(754,946,947,847,848,897)(761,953,954,854,855,892)(768,960,961,861,862,887)(775,967,968,868,869,882)(780,974,975,874,875,877)(785,980,972,879,880,872)(790,985,965,884,885,865)(795,990,958,889,890,858)(800,995,951,894,895,851)(805,1000,944,899,900,844)(809,1005,937,904,905,837)(812,1010,930,908,909,830)(823,1026,1027,922,923,1121)(825,1024,1025,925,921,1018)(828,915,1023,927,928,1019)(835,1037,1038,934,935,1014)(842,1044,1045,941,942,1009)(849,1051,1052,948,949,1004)(856,1058,1059,955,956,999)(863,1064,1065,962,963,994)(870,1069,1070,969,970,989)(876,1074,1075,976,977,984)(881,1079,1077,981,982,979)(886,1082,1072,986,987,973)(891,1085,1067,991,992,966)(896,1090,1062,996,997,959)(901,1095,1056,1001,1002,952)(906,1100,1049,1006,1007,945)(910,1105,1042,1011,1012,938)(913,1110,1035,1015,1016,931)(924,1083,1078,988,1029,1156)(926,1124,1125,1030,1028,1118)(929,1022,1123,1032,1033,1119)(936,1130,1131,1039,1040,1114)(943,1135,1136,1046,1047,1109)(950,1140,1141,1053,1054,1104)(957,1145,1146,1060,1061,1099)(964,1142,1149,1055,1066,1094)(971,1137,1152,1048,1071,1089)(978,1132,1155,1041,1076,1084)(983,1126,1158,1031,1080,1081)(993,1159,1153,1086,1087,1073)(998,1160,1150,1091,1092,1068)(1003,1164,1147,1096,1097,1063)(1008,1168,1143,1101,1102,1057)(1013,1172,1138,1106,1107,1050)(1017,1176,1133,1111,1112,1043)(1020,1180,1128,1115,1116,1036)(1034,1122,1157,1088,1127,1187)(1093,1201,1198,1161,1162,1154)(1098,1202,1196,1165,1166,1151)(1103,1203,1194,1169,1170,1148)(1108,1207,1192,1173,1174,1144)(1113,1211,1190,1177,1178,1139)(1117,1215,1188,1181,1182,1134)(1120,1200,1163,1184,1185,1129)(1167,1219,1220,1189,1186,1199)(1171,1228,1225,1204,1205,1197)(1175,1229,1223,1208,1209,1195)(1179,1230,1221,1212,1213,1193)(1183,1227,1206,1216,1217,1191)(1210,1234,1235,1222,1218,1226)(1214,1236,1233,1231,1232,1224);(1,2,4,5,3)(6,7,10,16,24,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,986,884,788,698,614,536,463,347,230,281,295,302,309,314,319,324,328,331,333,395,462,534,612,696,786,882,984,1081,1027,1123,1180,1176,1172,1168,1164,1160,1159,1029,1126,1132,1137,1142,1145,1140,1135,1130,1124,1026,1122,1162,1166,1170,1174,1178,1182,1185,1187,1156,1077,975,867,765,669,579,495,417,337,227,276,322,317,312,306,299,292,277,226,275,332,394,461,533,611,695,785,881,983,1030,925,824,729,640,557,480,409,344,285,232,185,144,109,80,57,38,25,9,13,18,28,42,61,84,113,148,189,236,289,348,413,484,561,644,733,828,929,1034,1086,991,889,793,703,619,540,412,283,340,354,361,368,374,379,384,389,393,396,398,466,539,617,701,791,887,989,1084,1125,920,1023,1110,1105,1100,1095,1090,1085,1082,1079,1074,1069,1064,1058,1051,1044,1037,1024,919,1022,1120,1186,1205,1209,1213,1217,1220,1198,1153,1072,972,871,772,676,586,502,424,352,278,180,223,263,258,253,246,239,224,179,222,273,329,390,456,528,606,690,780,876,978,1039,934,833,738,649,566,489,418,353,294,241,194,153,118,88,63,41,15,21,29,43,62,86,115,150,191,238,291,350,415,486,563,646,735,830,931,1036,1129,1161,1091,996,894,798,708,623,483,342,405,419,426,433,440,445,450,455,460,464,467,469,543,622,706,796,892,994,1089,1131,1025,819,916,1010,1005,1000,995,990,985,980,974,967,960,953,946,939,932,917,818,915,1020,1117,1183,1218,1232,1235,1225,1196,1150,1067,965,864,769,680,593,509,431,359,293,225,139,176,210,205,199,192,177,138,175,220,270,325,385,451,523,601,685,775,870,971,1046,941,840,745,656,573,496,425,360,301,248,201,159,123,92,60,23,34,44,46,66,91,121,157,198,245,298,357,422,493,570,653,742,837,938,1043,1134,1189,1165,1096,1001,899,803,712,560,407,476,490,497,504,511,517,522,527,532,537,541,544,546,626,711,801,897,999,1094,1136,1038,918,724,815,903,898,893,888,883,878,873,866,859,852,845,838,831,816,723,814,913,1017,1113,1179,1214,1233,1223,1194,1147,1062,958,857,762,673,590,513,438,366,300,240,178,104,135,163,158,151,136,103,134,173,217,266,320,380,446,518,596,679,768,863,964,1053,948,847,752,663,580,503,432,367,308,254,206,164,127,83,36,53,64,67,69,95,126,162,204,252,305,364,429,500,577,660,749,844,945,1050,1139,1191,1204,1169,1101,1006,904,807,643,478,553,567,574,581,588,595,600,605,610,615,620,624,627,629,715,806,902,1004,1099,1141,1045,933,817,635,720,802,797,792,787,782,777,771,764,757,750,743,736,721,634,719,812,910,1013,1108,1175,1210,1221,1192,1143,1056,951,850,755,666,583,506,435,370,307,247,193,137,75,100,122,116,101,74,99,132,170,213,261,315,375,441,512,589,672,761,856,957,1060,955,854,759,670,587,510,439,373,313,259,211,168,112,55,76,89,93,96,98,130,167,209,257,311,371,436,507,584,667,756,851,952,1057,1144,1193,1222,1208,1173,1106,1011,908,732,555,636,650,657,664,671,678,684,689,694,699,704,709,713,716,718,810,907,1009,1104,1146,1052,940,832,722,552,631,707,702,697,692,687,682,675,668,661,654,647,632,551,630,717,809,906,1008,1103,1171,1206,1190,1138,1049,944,843,748,659,576,499,428,363,304,251,200,152,102,52,71,87,72,51,70,97,129,166,208,256,310,369,434,505,582,665,754,849,950,1055,962,861,766,677,594,516,444,378,318,264,215,147,78,105,119,124,128,131,133,171,214,262,316,376,442,514,591,674,763,858,959,1063,1148,1195,1224,1231,1212,1177,1111,1015,827,638,725,739,746,753,760,767,774,779,784,789,794,799,804,808,811,813,911,1014,1109,1149,1059,947,839,737,633,475,548,618,613,608,603,598,592,585,578,571,564,549,474,547,628,714,805,901,1003,1098,1167,1188,1133,1042,937,836,741,652,569,492,421,356,297,244,197,156,117,73,33,48,49,32,47,68,94,125,161,203,250,303,362,427,498,575,658,747,842,943,1048,969,868,773,683,599,521,449,383,323,268,188,107,140,154,160,165,169,172,174,218,267,321,381,447,519,597,681,770,865,966,1068,1151,1197,1226,1236,1234,1216,1181,1115,928,727,820,834,841,848,855,862,869,875,880,885,890,895,900,905,909,912,914,1018,1114,1152,1065,954,846,744,648,550,404,471,535,530,525,520,515,508,501,494,487,472,403,470,545,625,710,800,896,998,1093,1163,1128,1035,930,829,734,645,562,485,414,349,290,237,190,149,114,85,50,20,31,19,30,45,65,90,120,155,196,243,296,355,420,491,568,651,740,835,936,1041,976,874,778,688,604,526,454,388,327,235,142,181,195,202,207,212,216,219,221,271,326,386,452,524,602,686,776,872,973,1073,1154,1199,1227,1230,1229,1228,1219,1184,1033,822,921,935,942,949,956,963,970,977,982,987,992,997,1002,1007,1012,1016,1019,1021,1118,1155,1070,961,853,751,655,565,473,339,400,458,453,448,443,437,430,423,416,401,338,399,468,542,621,705,795,891,993,1088,1032,927,826,731,642,559,482,411,346,287,234,187,146,111,82,59,40,27,12,11,17,26,39,58,81,110,145,186,233,286,345,410,481,558,641,730,825,926,1031,981,879,783,693,609,531,459,392,288,183,228,242,249,255,260,265,269,272,274,330,391,457,529,607,691,781,877,979,1078,1157,1200,1215,1211,1207,1203,1202,1201,1127,923,1028,1040,1047,1054,1061,1066,1071,1076,1080,1083,1087,1092,1097,1102,1107,1112,1116,1119,1121,1158,1075,968,860,758,662,572,488,402,280,335,387,382,377,372,365,358,351,336,279,334,397,465,538,616,700,790,886,988,922,821,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,14,8)Polycyclic presentation:
2,3,5,1231|c^a=c^4,d^a=d^1230,d^b=d^1261037 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36930, 17);