Finite Group DatabasebetaFinite Group Databasebeta| order | 32730 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1091⋊C5) |
| unambiguous semidirect product | (C1091⋊C5)⋊D3 |
| unambiguous semidirect product | D3273⋊C5 |
| Fitting, socle, derived, Hall | C3×C1091 ⋊ C2×C5 |
| Hall | C3×C1091⋊C5 ⋊ C2 |
| Hall | D3273 ⋊ C5 |
| Hall | C1091⋊C5 ⋊ D3 |
| Sylow | C1091 ⋊ C5×D3 |
| Sylow | C3 ⋊ D1091⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1091 | C2×C5 |
| LCS | G ▷ C3×C1091 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1094 |
| transitive degree | μt(G) = 3273 |
A minimal faithful permutation representation — 2 generators on 1094 points:
(2,3)(5,9,10,28,33,6,7,15,17,57)(8,22,23,90,95,12,13,44,46,174)(11,38,39,40,68,19,20,78,80,82)(14,51,52,208,213,25,26,106,108,405)(16,62,63,256,237,30,31,129,131,119)(18,72,73,69,70,35,36,74,34,81)(21,84,85,267,185,41,42,162,164,360)(24,100,101,379,142,48,49,195,197,283)(27,113,114,435,439,54,55,224,226,678)(29,123,124,58,239,59,60,244,120,121)(32,136,137,492,497,65,66,270,272,337)(37,148,149,186,187,75,76,294,96,362)(43,169,170,570,414,87,88,328,330,671)(45,179,180,404,460,92,93,347,349,238)(47,189,190,143,144,97,98,367,71,285)(50,202,203,399,361,103,104,393,395,618)(53,218,219,452,284,110,111,423,233,519)(56,231,232,510,511,116,117,450,279,601)(61,250,251,175,584,126,127,473,338,339)(64,265,266,83,314,133,134,363,159,160)(67,277,278,758,269,139,140,508,482,491)(77,300,301,415,416,151,152,535,214,673)(79,308,102,388,389,155,156,199,200,240)(86,324,247,248,154,166,167,125,384,306)(89,286,287,215,216,172,145,522,109,418)(91,341,342,122,463,176,177,589,241,242)(94,354,355,581,534,182,183,606,336,583)(99,373,374,364,365,192,193,634,188,622)(105,400,401,877,672,205,206,607,661,939)(107,410,411,692,736,210,211,428,669,461)(112,429,430,960,619,221,222,694,696,864)(115,444,445,596,520,228,229,718,346,686)(118,454,455,984,985,234,235,727,729,876)(128,479,480,406,255,253,254,705,446,481)(130,296,370,371,312,258,259,191,261,158)(132,369,291,292,322,262,263,147,245,165)(135,357,358,490,859,268,184,612,613,771)(138,502,503,643,555,274,275,783,785,725)(141,513,514,447,448,280,281,593,230,721)(146,297,298,385,386,289,150,257,198,246)(153,541,542,711,712,303,304,822,440,948)(157,551,220,691,528,310,311,426,427,585)(161,556,351,352,674,315,316,181,573,602)(163,560,504,665,738,319,320,608,787,462)(168,566,549,204,621,325,326,309,397,398)(171,577,578,592,307,332,333,343,453,547)(173,527,321,574,575,335,293,474,331,350)(178,594,515,340,524,344,345,800,586,587)(194,637,638,675,676,376,377,899,417,942)(196,647,648,652,740,381,382,916,915,464)(201,630,533,212,623,390,391,299,413,396)(207,553,494,495,419,403,313,271,537,679)(209,443,666,243,742,407,408,716,465,466)(217,681,317,624,625,420,421,557,366,412)(223,700,701,1083,940,432,433,967,968,1063)(225,708,709,627,628,437,438,475,368,737)(227,714,548,523,383,441,442,833,288,650)(236,730,731,1074,1075,457,458,987,909,1082)(249,749,559,359,861,470,471,318,615,616)(252,754,644,645,565,476,477,380,911,719)(260,760,761,776,777,483,484,868,498,620)(264,767,451,810,862,487,488,529,590,614)(273,779,739,329,746,499,500,626,468,572)(276,788,789,659,1049,505,506,1018,927,928)(282,802,803,1033,1034,516,517,919,827,872)(290,538,539,689,456,525,302,493,425,591)(295,811,812,704,747,530,531,597,786,469)(305,828,829,796,797,544,545,1036,512,1021)(323,532,636,631,632,562,563,375,372,387)(327,846,842,790,941,567,568,561,930,1020)(334,667,668,918,860,579,409,649,935,588)(348,867,848,765,660,598,599,780,486,521)(353,873,874,772,808,603,604,1056,1009,1010)(356,501,841,910,839,609,610,781,895,863)(378,904,905,971,972,640,641,1068,713,1081)(392,922,728,818,943,653,654,813,851,1019)(394,925,699,934,766,657,658,965,752,741)(402,882,884,885,550,662,663,617,726,834)(422,951,655,944,945,683,684,923,677,1029)(424,595,954,913,914,687,688,804,646,743)(431,744,745,436,552,698,467,912,706,707)(434,843,844,1054,680,703,564,999,901,946)(449,980,764,1087,1062,722,723,1002,1003,886)(459,881,989,906,1094,733,734,1027,1071,1072)(472,990,695,849,976,750,751,571,961,1045)(478,995,891,664,1050,755,756,656,931,932)(485,1000,1001,997,507,762,763,1059,759,791)(489,1005,1006,977,978,768,769,815,717,1057)(496,1013,1014,857,821,774,775,966,582,887)(509,824,974,835,837,793,794,715,936,554)(518,1024,898,1044,1093,805,806,1065,1066,1058)(526,825,826,735,605,809,543,792,720,875)(536,1028,957,958,991,816,817,693,1042,748)(540,784,697,917,892,820,795,962,963,1064)(546,1037,1038,988,1016,830,831,979,986,782)(558,893,894,852,853,840,633,710,576,1046)(569,920,921,1030,682,847,651,850,819,949)(580,937,938,970,1080,855,670,926,1079,1011)(600,1052,1053,856,1015,869,870,950,778,888)(611,1031,1061,929,1025,879,880,952,1077,956)(629,838,799,1012,1023,889,890,773,865,801)(635,992,823,982,993,896,897,866,964,753)(639,1051,690,883,814,902,903,955,732,1026)(642,1073,998,1022,871,907,908,807,798,1055)(685,1067,959,832,994,900,953,1069,973,1039)(702,1007,1008,1032,836,969,770,933,983,1040)(724,1047,1048,1076,947,981,854,1092,924,1078)(757,1035,1060,1004,1085,858,996,878,1088,1089)(845,1086,1070,1084,1041,1043,1017,1090,975,1091);(1,2,3)(4,5,8,14,27,56,118,236,459,735,718,479,310,155,20,40,83,161,317,558,841,1042,1091,1040,707,551,522,657,104,204,399,659,929,1053,949,681,948,980,842,560,572,850,433,702,970,1083,1073,1075,1014,965,698,429,695,170,331,576,854,972,684,952,1076,923,653,390,199,100,196,383,480,226,440,713,973,830,544,303,151,75,35,15,29,61,128,255,482,759,998,805,516,280,139,65,30,7,11,21,43,89,173,336,582,858,755,476,250,474,319,42,86,168,327,569,848,779,669,936,834,549,308,285,521,206,402,664,933,1078,946,679,418,678,721,559,85,165,323,564,845,1044,1071,904,1069,957,411,426,218,424,333,251,475,753,994,953,1074,907,640,376,192,97,44,91,178,346,442,715,975,905,1070,1088,1002,762,483,258,31,64,135,269,213,414,672,940,1062,884,615,357,182,92,13,24,50,105,207,404,665,934,750,470,247,123,245,468,88,171,334,580,856,776,492,691,833,547,306,39,81,158,313,554,838,1005,944,675,415,208,406,444,719,167,124,246,469,748,992,902,637,900,683,420,215,106,209,409,477,648,917,1037,638,901,1052,1092,1055,869,598,93,181,353,605,716,976,1085,1072,1001,661,930,788,774,494,66,138,276,507,416,674,941,1080,1009,613,164,265,413,210,26,53,112,223,434,704,783,744,465,241,120,17,34,71,145,288,524,808,1025,955,689,423,686,519,283,82,57,119,238,461,737,767,922,1038,977,408,441,224,436,326,248,379,643,910,374,636,373,635,898,938,925,211,412,670,756,918,960,828,903,761,867,1046,1021,1060,401,183,356,611,881,978,1056,1087,1093,985,355,608,502,784,793,140,279,512,798,676,943,1018,1007,768,487,262,129,257,437,55,115,230,449,724,982,962,694,809,525,289,36,74,147,293,528,584,859,1049,829,1012,688,594,450,725,160,121,240,462,739,190,369,630,892,803,668,647,438,710,568,471,645,912,893,631,370,189,368,629,891,203,396,622,888,996,1033,825,541,823,812,180,350,601,872,663,358,614,883,730,988,984,1000,1086,1016,610,499,270,493,773,281,515,801,1024,945,1028,1013,967,989,1026,563,259,294,529,117,233,453,726,983,542,824,961,873,727,530,76,150,299,534,497,95,185,361,619,514,578,341,590,863,316,242,464,741,765,371,256,388,652,913,644,101,198,387,651,751,914,742,624,364,186,90,175,340,588,398,266,362,620,887,820,538,300,536,818,352,570,508,792,932,616,342,591,454,728,811,760,999,731,880,1030,495,535,815,517,804,954,802,1006,301,537,819,1031,987,844,484,531,813,235,456,589,861,1050,826,278,330,573,851,816,152,302,540,821,777,187,363,621,860,586,338,46,96,188,366,466,743,990,920,632,385,195,380,646,915,200,131,261,486,766,740,463,556,839,810,177,343,593,864,618,360,174,337,583,533,297,148,295,455,604,849,794,822,1032,885,592,452,231,451,149,296,532,814,1027,701,775,817,1029,806,1023,800,513,799,539,137,273,501,782,1017,763,729,986,458,732,862,612,882,1034,511,575,347,597,866,304,543,827,1035,1015,365,623,393,656,889,627,98,191,372,633,745,911,749,846,894,708,382,649,919,1064,391,263,367,626,738,389,239,314,555,232,345,595,865,1036,928,771,339,585,527,291,73,72,146,290,526,430,697,964,981,722,447,228,113,225,298,63,132,264,489,770,789,1019,942,1022,796,510,277,509,795,275,504,606,876,1058,1003,874,1057,734,879,609,354,607,878,797,853,599,868,1051,545,696,935,995,937,625,410,658,926,1065,896,193,375,634,895,785,197,384,566,706,114,227,443,717,979,654,488,628,736,460,237,33,68,142,284,520,219,425,690,956,1010,587,523,286,143,69,28,58,122,243,467,503,786,703,432,221,110,51,107,212,134,267,490,772,1011,1020,602,673,791,505,274,136,271,496,506,790,877,1059,1094,1089,1045,666,875,603,351,179,348,600,871,1048,870,1054,899,831,963,916,754,667,407,52,109,217,422,685,377,639,897,991,747,386,244,324,565,229,446,108,214,417,677,769,890,837,553,312,70,78,154,307,548,427,272,498,778,855,579,332,169,329,292,60,125,249,472,752,787,349,403,205,103,48,22,45,94,184,359,617,886,1063,939,671,405,491,268,133,62,130,260,485,764,1004,1090,1068,1084,974,714,596,344,176,23,47,99,194,378,642,909,1067,1039,993,709,473,577,687,111,220,428,693,959,641,906,1066,1043,843,562,322,162,318,448,439,216,419,680,947,1008,931,662,400,660,144,156,309,550,835,692,500,780,847,567,325,166,84,163,321,127,252,478,757,857,581,335,172,87,41,19,9,16,32,67,141,282,518,807,997,758,481,253,126,59,10,18,37,77,153,305,546,832,971,711,435,705,650,381,49,102,201,392,655,924,1061,951,1081,1047,852,574,328,571,222,431,699,966,1082,908,968,1079,969,700,921,746,320,561,723,712,421,682,950,1077,927,395,397,202,394,287,311,552,836,1041,958,781,840,557,315,159,80,38,79,157,254,445,720,733,457,234,116,54,25,12,6)Polycyclic presentation:
2,3,5,1091|b^a=b^2,d^a=d^1090,d^c=d^93342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32730, 12);