Finite Group DatabasebetaFinite Group Databasebeta| order | 20106 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #23758409⋊C2 |
| Fitting, socle, derived, Sylow | C1117 ⋊ C2×C9 |
| Hall | #23758409 ⋊ C2 |
| Hall | D1117 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C1117 | C2×C9 |
| prime radical | D1117⋊C3 | C3 |
| cosocle | C1117⋊C3 | C2×C3 |
| LCS | G ▷ C1117 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1117 |
| transitive degree | μt(G) = 1117 |
A minimal faithful permutation representation — 2 generators on 1117 points:
(2,19,53,8,24,18,12,7,22,3,10,74,5,30,9,13,4,43)(6,80,199,33,86,79,37,32,84,14,35,381,16,190,34,82,15,188)(11,152,129,64,158,151,68,63,156,26,66,286,28,323,65,154,27,250)(17,225,61,95,145,224,60,94,25,39,97,144,41,310,96,143,40,59)(20,257,111,113,176,256,117,112,136,46,115,177,48,73,114,146,47,297)(21,221,372,92,182,278,76,123,222,50,93,183,38,77,124,31,51,451)(23,118,141,131,58,288,135,130,292,54,133,306,56,140,132,290,55,265)(29,354,402,170,357,258,174,169,356,70,172,489,72,647,171,355,71,500)(36,426,287,211,432,425,215,210,430,88,213,552,90,744,212,428,89,742)(42,185,335,235,186,396,237,197,159,100,78,196,102,83,236,471,101,334)(44,165,309,246,167,231,249,180,166,105,69,179,107,75,247,227,106,348)(45,346,296,164,390,331,255,192,243,109,193,244,110,294,157,134,81,104)(49,517,725,268,521,516,272,267,410,119,270,684,121,681,269,518,120,724)(52,345,562,232,469,344,234,163,293,126,233,650,99,358,283,195,127,561)(57,391,284,300,205,238,304,299,311,137,302,128,139,415,301,569,138,474)(62,319,404,315,320,584,317,314,200,147,150,717,149,588,316,585,148,403)(67,603,307,338,333,620,342,337,624,160,340,576,162,613,339,622,161,926)(85,530,153,407,142,601,366,406,602,201,408,325,203,308,326,178,202,615)(87,548,230,418,466,637,229,417,98,206,420,465,208,625,419,464,207,228)(91,657,488,363,772,175,446,443,328,217,444,248,219,670,362,658,218,364)(103,645,741,476,592,813,479,475,433,239,353,911,241,586,477,814,240,755)(108,439,768,492,441,830,495,491,440,251,216,692,253,386,493,804,252,767)(116,630,578,507,632,844,510,506,631,261,343,903,263,596,508,570,262,955)(122,870,352,532,643,581,351,531,168,274,534,312,276,963,533,642,275,350)(125,653,566,360,882,652,546,359,753,173,544,577,282,754,431,654,281,490)(155,656,731,361,575,894,305,608,414,329,505,702,260,565,609,295,330,730)(181,685,718,383,942,189,660,666,649,368,619,687,370,661,382,365,369,384)(184,915,463,594,983,322,677,674,752,374,675,802,376,887,593,547,375,496)(187,460,805,226,812,908,473,639,468,378,583,956,379,722,640,461,349,797)(191,766,824,438,901,825,571,694,484,387,696,485,389,572,695,303,388,245)(194,324,635,697,629,953,698,626,920,392,597,922,393,599,627,921,341,598)(198,1001,1100,709,976,499,703,708,1002,398,711,977,400,659,710,395,399,838)(204,749,733,726,416,880,728,573,750,411,429,893,413,732,727,900,412,1021)(209,481,721,740,482,881,720,739,405,422,242,1009,424,819,540,1008,423,719)(214,1023,327,757,614,928,760,756,929,434,758,604,436,936,605,987,435,1091)(220,891,313,555,950,289,781,671,527,448,672,529,449,854,554,528,373,273)(223,497,467,789,498,579,792,788,487,454,254,803,456,836,790,678,455,827)(259,667,787,526,668,982,587,841,453,502,371,924,318,795,842,458,503,600)(264,946,847,849,509,782,852,848,975,511,655,835,513,846,850,560,512,1046)(266,636,826,347,567,853,298,541,486,459,543,514,515,723,542,409,279,280)(271,1029,904,863,1031,1073,866,862,1030,522,761,945,524,1017,864,1074,523,1113)(277,1065,706,877,808,707,879,876,633,537,829,397,539,1062,878,1081,538,957)(285,840,377,817,933,839,818,501,883,549,607,679,480,686,680,809,550,969)(291,452,1071,574,886,1035,611,851,919,557,785,935,559,612,765,332,558,786)(321,1014,618,914,939,1000,617,735,336,590,737,938,591,743,736,937,421,616)(367,860,1016,520,823,427,923,610,973,663,958,1018,665,1012,745,729,664,746)(380,810,1050,472,856,995,816,858,984,682,796,857,683,519,964,811,644,1063)(385,1044,1109,934,993,1105,991,988,1045,688,780,994,690,693,989,1093,689,1112)(394,648,907,996,974,815,773,916,662,700,959,582,701,968,478,861,634,967)(401,896,913,1004,1034,895,885,563,843,713,1005,589,715,906,910,580,714,504)(437,952,892,874,971,536,943,1032,855,762,1033,556,764,1096,641,1080,763,875)(442,621,873,734,1025,783,872,1039,535,770,944,1069,676,930,450,1013,771,832)(445,1059,606,1040,1022,1075,1003,986,1085,774,888,931,776,1047,932,1098,775,1116)(447,1019,978,748,834,1041,494,868,981,777,1043,867,779,525,1042,833,778,899)(457,1020,1010,1007,990,909,1051,1006,970,716,1049,1108,794,1107,1048,1103,793,1082)(462,1104,1076,1054,865,1066,747,1053,1099,798,1056,1101,800,979,1055,704,799,1110)(470,568,638,553,751,960,951,712,828,807,898,961,673,699,897,890,564,845)(483,869,628,948,917,623,972,837,651,821,1015,918,822,595,947,1095,738,949)(545,927,889,1057,954,820,1058,1026,1064,884,1028,551,801,784,1027,1102,759,985)(646,1089,1061,912,1094,1114,940,1036,806,965,1038,1106,966,1024,1037,1090,769,1060)(669,1086,941,902,1067,705,992,962,925,859,905,1068,980,831,997,691,791,998)(871,999,1072,1083,1092,1011,1088,1117,1097,1077,1087,1115,1079,1070,1111,1052,1078,1084);(1,2,6,17,42,103,242,483,823,618,941,1091,1030,989,1106,1063,915,719,403,188,257,137,54,18,44,108,254,499,736,878,612,654,974,1005,1074,1080,685,616,334,152,324,600,451,730,411,201,79,189,385,691,895,317,533,383,686,687,717,422,206,80,191,302,346,635,725,199,402,488,180,224,458,796,1052,1048,316,269,520,861,657,870,1028,340,601,925,838,1071,1019,1096,562,647,846,506,396,705,999,1086,790,145,114,260,355,356,649,970,539,880,1084,805,955,1085,1117,1100,661,979,862,813,682,378,185,76,12,8,21,52,128,285,551,543,565,706,335,615,771,1038,1076,1107,988,1013,1008,724,407,507,611,331,151,322,595,919,369,664,608,77,184,377,681,987,1104,860,957,630,338,623,950,826,511,261,112,258,501,841,694,148,240,478,795,991,1108,1009,770,439,211,429,751,697,663,366,176,70,26,7,20,49,122,277,540,424,627,921,1042,1040,1069,758,1025,928,1092,498,837,735,417,292,127,160,63,153,327,606,897,419,515,110,154,328,607,934,1077,276,536,639,348,638,962,788,1035,762,434,210,427,747,703,234,37,92,33,83,198,401,716,454,221,452,632,158,333,614,768,847,833,491,464,598,923,726,849,825,652,425,743,868,764,799,822,567,789,1047,882,744,1017,1095,505,558,848,233,470,808,485,559,569,72,175,365,662,617,940,591,572,902,1079,1081,887,1044,777,363,659,978,996,728,510,139,305,255,130,289,556,893,375,676,208,294,564,708,528,440,769,864,1075,1067,253,496,742,624,951,918,626,337,621,945,903,179,308,140,30,75,181,371,669,856,121,273,530,230,467,804,1015,834,522,267,281,545,885,474,60,101,232,95,226,462,801,1059,677,360,655,342,629,954,783,448,217,88,32,81,194,394,702,542,538,339,625,952,524,436,761,482,820,1007,1058,1018,911,739,541,51,125,169,141,307,577,244,290,329,28,69,168,353,646,966,320,589,858,642,750,857,1006,850,555,774,443,609,852,557,56,136,298,406,722,1011,964,993,441,86,205,416,658,975,1099,1109,508,845,827,107,250,490,829,672,982,1026,756,1024,1112,640,460,529,117,144,312,582,696,843,502,48,118,266,97,227,463,737,763,1034,680,986,1103,818,237,473,709,301,235,472,746,544,883,959,172,358,651,787,963,1098,1001,412,571,299,568,899,772,183,373,673,132,293,563,270,213,433,241,480,715,800,811,814,1064,792,594,916,283,548,552,286,74,177,109,10,25,62,150,321,592,149,318,518,855,1072,1114,1055,493,832,531,124,280,484,243,22,40,98,231,468,806,1049,779,215,438,389,428,748,1020,1060,797,247,228,94,43,104,245,486,278,275,535,830,1066,1037,1115,875,272,526,315,586,590,319,147,59,19,45,111,53,129,287,420,344,634,376,678,985,479,816,1054,1004,817,476,755,426,517,714,561,288,553,671,372,670,981,898,138,303,573,711,1003,643,924,949,469,354,648,969,653,973,683,102,238,400,461,471,809,1051,775,839,906,1032,1014,802,249,225,459,133,113,259,504,766,907,352,61,146,313,583,908,1045,1094,435,759,842,891,1065,753,156,246,487,828,844,994,1110,1046,446,732,415,190,386,693,995,1111,812,202,409,297,131,291,560,894,218,445,449,782,793,1050,1021,351,644,913,588,912,965,645,350,165,64,155,135,296,566,576,306,71,173,123,279,423,741,1016,1102,794,1027,819,1029,757,863,1033,466,620,876,853,731,700,392,192,15,36,91,220,450,784,599,622,946,282,547,888,1057,798,379,41,99,197,143,311,580,884,359,120,271,525,869,495,803,465,408,527,268,519,859,667,368,167,24,58,142,309,578,904,944,161,341,628,890,926,430,752,492,831,932,1073,1036,767,781,399,712,390,418,734,674,733,892,554,55,134,295,300,570,900,701,867,976,219,447,780,983,879,1083,980,901,914,1090,937,967,660,362,170,304,574,824,1062,807,345,512,851,656,972,1031,432,754,1022,456,723,948,1053,874,778,939,212,431,695,513,413,729,920,229,252,494,835,692,936,613,323,596,785,93,223,457,713,398,186,16,38,82,195,395,704,745,89,214,437,765,455,791,961,166,349,641,532,871,690,840,364,68,164,347,637,960,931,604,325,27,67,163,265,207,421,738,836,1070,605,930,1023,873,776,1041,698,953,740,881,537,274,119,46,4,11,29,73,178,367,393,699,749,90,216,442,721,1010,1093,668,815,475,314,388,503,550,171,47,116,264,514,854,947,162,343,633,958,1056,760,521,679,374,182,330,610,666,786,917,593,65,157,332,263,203,410,720,872,689,990,1101,1089,1039,602,196,397,575,636,889,549,284,126,50,5,13,31,78,187,380,477,523,865,942,977,1078,1116,631,956,1097,727,877,1082,384,500,174,361,256,310,579,905,1087,997,236,262,509,357,650,971,1043,935,1002,998,326,603,927,534,444,773,665,516,584,909,1088,810,587,96,106,248,489,381,684,922,193,391,387,35,87,209,404,718,933,370,581,585,910,992,688,382,34,85,204,414,222,453,597,66,159,336,619,943,866,896,968,546,886,707,1000,710,497,251,105,9,23,57,115,84,200,405,675,984,1061,1105,1113,929,1068,938,1012,821,481,239,100,39,14,3)Polycyclic presentation:
2,3^2,1117|b^3=c,d^a=d^1116,d^b=d^928,d^c=d^99680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20106, 8);