Finite Group DatabasebetaFinite Group Databasebeta| order | 34530 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1151⋊C5) |
| unambiguous semidirect product | (C1151⋊C5)⋊D3 |
| unambiguous semidirect product | D3453⋊C5 |
| Fitting, socle, derived, Hall | C3×C1151 ⋊ C2×C5 |
| Hall | C3×C1151⋊C5 ⋊ C2 |
| Hall | D3453 ⋊ C5 |
| Hall | C1151⋊C5 ⋊ D3 |
| Sylow | C1151 ⋊ C5×D3 |
| Sylow | C3 ⋊ D1151⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1151 | C2×C5 |
| LCS | G ▷ C3×C1151 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1154 |
| transitive degree | μt(G) = 3453 |
A minimal faithful permutation representation — 2 generators on 1154 points:
(2,3)(5,9,10,28,33,6,7,15,17,57)(8,22,23,90,95,12,13,44,46,175)(11,38,39,40,68,19,20,78,80,82)(14,51,52,209,214,25,26,106,108,403)(16,62,63,254,235,30,31,129,131,119)(18,72,73,69,70,35,36,74,34,81)(21,84,85,265,186,41,42,162,164,357)(24,100,101,376,142,48,49,196,198,282)(27,113,114,428,432,54,55,224,226,703)(29,123,124,58,237,59,60,242,120,121)(32,136,137,490,495,65,66,269,271,341)(37,148,149,187,188,75,76,293,96,359)(43,169,170,569,407,87,88,328,330,666)(45,180,181,604,450,92,93,350,352,236)(47,190,191,143,144,97,98,364,71,284)(50,203,204,396,358,103,104,390,392,213)(53,219,220,555,283,110,111,417,317,512)(56,229,230,720,723,116,117,439,441,897)(61,248,249,176,591,126,127,463,342,343)(64,263,264,83,314,133,134,360,159,160)(67,276,277,819,786,139,140,505,507,658)(77,299,300,408,409,151,152,531,215,668)(79,307,102,385,386,155,156,200,201,238)(86,324,245,246,154,166,167,125,381,305)(89,335,336,418,443,172,173,580,231,686)(91,345,346,122,453,177,178,594,239,240)(94,275,355,815,648,183,184,503,611,590)(99,370,371,361,362,193,194,627,189,615)(105,397,398,946,667,206,207,649,651,955)(107,112,406,423,738,211,212,222,664,451)(109,411,412,285,286,216,217,673,145,514)(115,405,437,455,456,210,228,662,241,744)(118,444,445,1022,1025,232,233,729,731,1102)(128,469,470,404,783,251,252,761,659,660)(130,295,367,368,312,256,257,192,259,158)(132,366,290,291,322,260,261,147,243,165)(135,485,486,484,356,266,267,787,785,185)(138,500,501,633,553,273,274,808,810,871)(141,179,510,599,344,279,280,348,402,593)(146,296,297,382,383,288,150,255,199,244)(153,537,471,707,708,302,303,764,433,685)(157,546,221,689,644,309,310,420,421,592)(161,554,278,354,669,315,316,182,509,610)(163,558,502,902,740,319,320,812,813,452)(168,566,544,205,614,325,326,308,394,395)(171,576,577,597,306,332,333,347,556,542)(174,584,585,881,896,338,339,888,860,600)(195,630,631,670,671,373,374,928,410,959)(197,636,637,642,742,378,379,934,581,454)(202,623,528,529,616,387,388,298,483,393)(208,655,550,795,728,400,401,866,867,1020)(218,679,680,617,618,414,415,972,363,915)(223,697,698,1055,956,425,426,993,779,832)(225,334,706,882,937,430,431,578,638,739)(227,710,711,515,516,434,435,1006,287,833)(234,734,735,801,914,447,448,919,1030,727)(247,752,557,487,892,460,461,318,790,791)(250,757,634,635,565,466,467,377,932,858)(253,766,746,704,656,472,473,1036,864,721)(258,774,775,799,800,476,477,906,496,613)(262,780,781,841,893,480,481,523,595,789)(268,793,474,877,1049,488,489,768,769,1084)(270,533,676,677,551,492,493,413,607,313)(272,802,803,747,748,497,498,620,458,873)(281,380,830,709,982,375,511,639,1003,1004)(289,534,535,687,568,518,301,491,419,596)(292,464,331,353,675,520,521,573,574,608)(294,843,844,903,750,524,525,605,811,459)(304,859,794,657,895,539,540,899,724,598)(311,852,438,1015,482,548,549,717,718,782)(321,845,828,1070,389,560,561,1079,960,643)(323,527,629,624,625,562,563,372,369,384)(327,846,847,816,958,567,526,559,948,1071)(329,536,809,693,936,571,572,856,987,741)(337,705,586,745,543,429,583,663,457,862)(340,661,890,601,602,587,588,870,349,898)(351,737,905,777,778,449,606,804,479,513)(365,921,771,772,755,621,622,475,935,465)(391,944,696,1103,1034,646,647,990,991,743)(399,900,901,798,545,653,603,792,872,863)(416,977,978,961,962,682,683,1128,672,1122)(422,983,984,1056,916,690,691,922,784,1111)(424,989,751,532,547,694,695,933,853,854)(427,999,1000,1146,1021,700,701,1136,1062,1110)(436,1011,979,917,918,713,714,1130,619,1112)(440,654,1019,758,759,645,722,823,468,1033)(442,589,967,834,835,725,726,891,517,1061)(446,954,522,762,582,665,733,838,839,886)(462,837,538,820,716,753,519,570,767,1013)(478,861,1054,1026,1059,541,776,1074,824,957)(494,865,996,997,952,796,797,699,910,913)(499,1064,963,1100,715,805,806,926,894,951)(504,1072,1073,1108,1083,817,818,1076,907,831)(506,765,1008,1009,869,821,822,712,953,552)(508,1075,1069,1038,1039,825,826,964,749,1089)(530,788,652,814,968,849,850,949,950,1042)(564,1053,929,969,970,874,875,1029,678,1068)(575,1032,974,923,924,878,879,1065,626,1114)(579,1099,730,998,1121,883,884,945,1024,1085)(609,1109,1088,1081,702,908,909,842,829,770)(612,995,992,1040,1041,911,912,920,754,857)(628,947,1104,1105,1035,650,927,904,988,756)(632,1107,688,1058,848,930,931,981,885,1090)(640,1119,1014,1043,1044,939,940,1137,760,1097)(641,1098,855,975,976,941,942,1092,681,1063)(674,1067,1050,1051,1045,965,966,773,1082,763)(684,1096,1131,851,1060,868,980,1028,1005,1091)(692,1133,1134,1151,1123,985,986,1116,1080,1135)(719,1138,807,1145,1113,1016,1017,1066,1057,1117)(732,1120,1141,1012,1086,943,1027,1047,836,1132)(736,1129,925,1124,887,994,1031,971,1115,880)(827,1148,1149,1152,1018,1077,1078,1125,1101,1139)(840,1048,889,1002,1118,1010,1087,876,1094,938)(973,1150,1147,1153,1093,1126,1127,1106,1142,1046)(1001,1140,1144,1143,1095,1023,1052,1154,1037,1007);(1,2,3)(4,5,8,14,27,56,118,234,449,93,182,276,506,823,791,346,596,567,325,166,84,163,321,481,622,922,985,690,420,219,418,229,440,204,393,615,914,725,439,721,655,621,98,192,369,626,925,969,676,411,674,967,1118,830,1082,1124,1105,987,664,953,769,330,509,828,1080,1149,680,974,1064,1133,850,267,487,792,1059,409,669,958,1121,516,549,300,533,855,714,1012,834,515,285,143,69,28,58,122,241,457,746,1037,942,748,320,559,861,1094,578,467,637,936,740,386,237,314,553,316,240,454,743,1035,755,463,576,881,859,557,85,165,323,564,876,888,837,518,288,36,74,147,292,522,840,1008,710,1007,797,854,546,514,832,955,666,403,658,785,164,263,483,784,1057,1000,1127,1148,1077,825,505,820,766,1048,695,501,811,1068,1144,940,1054,1072,1126,415,681,735,917,617,361,187,90,176,344,432,443,728,1021,618,916,647,945,233,446,222,424,696,992,803,191,366,623,571,88,171,334,579,885,893,787,900,601,280,177,23,47,99,195,375,279,139,65,30,7,11,21,43,89,174,340,589,817,184,273,136,270,494,798,597,555,315,159,80,38,79,157,311,550,868,682,414,216,106,210,113,225,395,264,359,613,913,722,653,397,650,194,372,627,926,961,670,408,209,404,661,882,636,935,1114,1153,1131,954,1006,908,574,350,605,904,1106,976,905,802,983,1091,489,794,898,1102,671,960,611,910,835,867,535,137,272,499,807,1067,1109,1038,747,291,60,125,247,462,437,716,1014,1120,1039,561,843,774,1053,884,759,937,738,450,235,33,68,142,283,456,336,582,887,878,573,328,570,537,858,167,124,244,459,751,932,752,846,524,76,150,298,530,851,1003,707,428,704,863,544,307,284,513,831,1084,660,782,480,260,129,255,475,773,1052,1073,1075,1138,1141,822,764,469,762,991,813,352,607,907,651,948,355,612,683,979,1031,1128,1152,1100,841,178,347,598,708,220,419,688,918,1113,631,929,448,736,426,699,998,1055,777,368,254,385,642,173,337,114,227,436,715,949,649,939,511,378,49,102,202,389,644,591,356,183,92,13,24,50,105,208,402,657,720,488,266,133,62,130,258,478,539,302,151,75,35,15,29,61,128,253,474,770,700,425,212,110,51,107,213,357,175,341,590,392,394,203,391,374,632,927,1116,1139,1016,717,405,663,584,634,101,199,384,641,943,729,1023,999,879,606,906,1107,1146,1030,799,490,689,982,723,896,510,829,962,1123,912,993,1086,1096,856,274,502,814,1069,412,675,968,1040,521,127,250,468,760,718,507,824,1022,1088,781,149,295,527,848,1044,890,869,551,312,70,78,154,306,543,583,470,763,1046,1041,572,534,299,532,326,246,376,633,810,198,381,566,853,152,301,536,857,1093,1045,761,705,862,542,305,39,81,158,313,552,870,1097,1090,563,257,293,523,842,731,1026,819,1015,1043,758,466,248,464,754,1042,608,673,964,950,812,500,809,980,1132,698,995,1135,1122,1081,348,600,897,1004,421,271,496,801,1062,931,775,737,1032,701,1001,445,732,941,625,382,196,377,339,586,228,438,719,1018,1134,947,930,630,646,104,205,396,648,495,95,186,358,211,26,53,112,223,427,702,768,472,251,126,59,10,18,37,77,153,304,541,476,256,31,64,135,268,441,724,599,400,206,103,48,22,45,94,185,343,592,643,387,200,100,197,380,640,398,652,951,713,434,224,429,335,581,201,131,259,479,779,1024,996,697,994,734,1029,928,1117,1112,981,687,417,685,895,577,345,595,894,1101,978,1129,1130,977,911,503,816,946,1108,677,604,902,1103,839,252,471,765,1047,1017,826,1076,1140,1050,771,297,63,132,262,482,783,1049,1083,778,144,156,308,545,864,226,433,709,1005,849,528,296,148,294,526,461,635,933,750,383,242,324,565,303,538,170,331,575,880,886,580,744,512,282,82,57,119,236,451,739,1033,1099,875,477,525,845,1089,1027,1137,1013,662,753,460,245,123,243,458,749,909,966,1066,805,497,269,491,795,1061,997,815,1070,959,1025,602,899,793,1060,691,498,804,1063,1147,1104,844,181,353,609,711,733,1028,1142,924,772,379,638,588,659,108,215,410,672,963,371,629,370,628,207,399,654,952,800,188,360,614,430,55,115,52,109,218,416,684,866,548,309,155,20,40,83,161,317,556,872,796,492,66,138,275,504,726,587,338,172,87,41,19,9,16,32,67,141,281,373,193,97,44,91,179,349,603,486,789,1058,883,431,332,169,329,388,261,364,620,920,990,694,406,665,444,730,944,1111,915,1110,1020,686,703,593,342,46,96,189,363,619,919,975,679,973,818,1074,1119,1154,970,903,808,989,1087,473,767,277,508,827,1078,1150,1136,1145,1056,529,134,265,484,786,214,407,667,956,286,310,547,865,1095,435,712,1010,838,520,290,73,72,146,289,519,585,889,874,562,322,162,318,540,860,333,249,465,756,1034,742,453,554,871,160,121,238,452,741,934,757,706,1002,776,847,558,873,1098,1143,1036,745,455,239,120,17,34,71,145,287,517,836,1011,1092,493,531,852,833,1085,1071,610,668,957,901,790,485,788,986,806,1065,972,1125,1151,1079,354,569,877,1009,423,693,988,1115,1051,639,938,891,965,217,413,678,971,923,624,367,190,365,401,656,230,442,727,362,616,390,645,117,231,111,221,422,692,984,921,780,560,319,42,86,168,327,568,594,892,1019,821,140,278,180,351,447,232,116,54,25,12,6)Polycyclic presentation:
2,3,5,1151|b^a=b^2,d^a=d^1150,d^c=d^224360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34530, 12);