C₂×#23758409 #34117332

order	20106
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂×#23758409

Splittings

Fitting, socle, Hall	C₂×C₁₁₁₇ ⋊ C₉
derived, Sylow	C₁₁₁₇ ⋊ C₂×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#23758409
commutator, hypocenter	C₁₁₁₇	C₂×C₉
Frattini	C₁	G
Fitting, socle	C₂×C₁₁₁₇	C₉
prime radical	C₂×C₁₁₁₇⋊C₃	C₃
cosocle	C₁₁₁₇⋊C₃	C₂×C₃
LCS	G ▷ C₁₁₁₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1119
transitive degree	μ_t(G) = 2234

A minimal faithful permutation representation — 2 generators on 1119 points:

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

Conjugacy classes & character table

266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20106, 9);

C2×#23758409 #34117332