Finite Group DatabasebetaFinite Group Databasebeta| order | 13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #17985008⋊C3 |
| Fitting, socle, derived, Sylow | C1117 ⋊ C3×C4 |
| Hall | #17985008 ⋊ C3 |
| Hall | C1117⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C1117 | C3×C4 |
| prime radical | D1117⋊C3 | C2 |
| cosocle | D1117 | 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:
(2,7,12,8,4,17,5,18,10,3,11,26)(6,28,69,29,14,61,15,62,31,13,32,130)(9,46,104,20,22,94,23,44,47,21,48,99)(16,73,161,74,35,153,36,154,76,34,77,310)(19,49,189,90,41,107,42,108,92,40,93,57)(24,110,224,64,51,222,52,137,111,50,112,278)(25,72,27,118,33,124,55,148,59,54,70,58)(30,139,284,45,66,282,67,101,140,65,141,205)(37,167,342,168,80,335,81,336,170,79,171,615)(38,88,100,177,84,105,85,39,179,83,180,87)(43,142,391,195,96,287,97,288,197,95,198,265)(53,231,449,156,115,447,116,317,232,114,233,585)(56,164,147,244,78,332,121,333,165,120,166,297)(60,258,150,259,127,328,128,149,162,126,71,300)(63,230,388,250,134,251,135,445,249,133,273,123)(68,290,340,138,144,339,145,280,169,143,291,208)(75,319,590,102,158,589,159,207,320,157,321,409)(82,348,636,349,174,630,175,631,351,173,352,955)(86,270,408,362,182,285,183,132,364,181,365,269)(89,371,266,372,186,131,187,279,373,185,374,267)(91,227,212,239,113,442,191,443,228,190,229,416)(98,322,694,395,201,592,202,593,397,200,398,572)(103,393,379,255,199,256,210,485,254,209,394,125)(106,420,178,421,215,253,216,437,359,214,225,358)(109,425,383,307,219,308,220,382,306,218,193,152)(117,456,776,338,236,774,237,621,457,235,458,947)(119,407,385,361,241,194,242,206,465,240,466,386)(122,345,327,475,172,627,247,628,346,246,347,601)(129,496,330,497,262,626,263,329,343,261,163,604)(136,514,839,515,275,823,276,643,517,274,518,501)(146,546,699,318,294,401,295,587,400,293,547,203)(151,562,298,536,303,827,304,864,554,302,563,553)(155,579,691,481,314,482,315,805,480,313,581,252)(160,595,799,281,324,798,325,525,476,323,596,855)(176,641,614,642,355,959,356,632,644,354,645,940)(184,577,921,657,368,591,369,312,659,367,660,576)(188,666,573,667,376,311,377,586,668,375,669,574)(192,452,436,471,234,499,380,637,453,344,454,264)(196,543,531,464,292,662,390,788,544,389,545,370)(204,703,366,704,404,299,405,418,656,403,213,556)(211,697,686,578,399,831,414,908,506,413,698,552)(217,606,502,607,422,609,423,937,726,331,727,555)(221,735,688,736,428,889,429,687,695,427,392,558)(223,634,685,487,433,488,434,350,486,432,411,257)(226,750,360,751,439,491,440,768,649,438,450,648)(238,783,583,633,461,582,462,824,316,460,784,913)(243,791,741,664,468,431,469,524,498,467,792,742)(245,712,857,651,473,527,474,410,658,472,796,711)(248,639,625,800,353,930,478,957,600,477,640,949)(260,283,858,674,493,529,494,530,813,492,528,710)(268,508,854,826,504,624,505,271,828,503,829,507)(272,772,417,490,455,1001,511,811,720,510,773,719)(277,841,1066,842,520,834,521,1033,844,519,845,905)(286,526,363,862,534,484,535,623,653,533,341,652)(289,866,836,569,538,570,539,835,568,537,513,309)(296,881,1094,622,549,1085,550,948,882,548,883,904)(301,891,682,655,560,387,561,713,892,559,893,683)(305,895,602,896,565,1080,566,747,898,564,899,931)(326,900,939,588,567,717,599,861,716,598,901,415)(334,446,821,678,612,679,613,762,638,611,500,384)(337,943,647,807,618,357,619,1052,806,617,944,483)(378,516,884,859,671,616,672,838,551,670,837,942)(381,779,767,797,459,804,676,941,780,675,781,1004)(396,927,920,789,597,1007,693,1041,928,692,929,724)(402,489,448,786,702,808,463,809,765,701,787,764)(406,994,933,995,706,603,707,752,973,705,665,934)(412,856,681,725,715,757,605,956,981,714,635,684)(419,938,709,926,661,969,723,610,970,608,971,932)(424,843,906,1009,729,952,730,1068,1010,728,1011,1023)(426,444,690,731,733,759,734,760,972,732,758,663)(430,470,988,1016,739,793,740,794,919,738,696,936)(435,922,1067,918,745,986,746,1051,888,744,1020,886)(441,1026,825,877,754,1065,755,917,1027,753,1028,1063)(451,1035,650,1019,770,819,771,1038,965,769,777,964)(479,961,954,1050,646,953,803,680,629,802,962,979)(495,541,1049,983,815,872,816,873,801,814,871,1077)(509,984,907,985,766,575,832,1096,1031,689,894,763)(512,916,748,865,782,1098,833,584,1022,778,1039,915)(522,968,1029,1070,847,756,848,1097,1071,846,1072,880)(523,1076,830,1059,851,557,852,860,1064,850,532,887)(540,1053,1037,1084,810,775,869,963,1054,868,840,890)(542,1042,654,1088,875,812,876,990,967,874,700,966)(571,594,978,1040,902,677,903,924,1048,785,923,912)(580,1099,721,818,910,1110,911,1058,1003,909,1100,1002)(620,975,1083,1106,673,867,946,987,976,945,977,1062)(708,743,1095,982,997,951,998,1018,885,996,974,991)(718,817,879,935,992,1055,1000,1056,950,999,853,1092)(722,1081,1024,993,897,749,1005,863,820,925,790,1025)(737,878,1078,1012,1014,1091,1015,1057,1093,1013,1090,1008)(761,989,1082,1061,1030,980,958,1105,1060,960,1107,822)(795,914,1075,1079,1045,1036,1046,1102,1047,1044,1101,1108)(849,1086,1115,1087,1073,1116,1074,1069,1104,870,1113,1032)(1006,1117,1021,1109,1111,1034,1112,1089,1017,1103,1043,1114);(1,2,6,16,37,82,176,357,482,251,107,17,38,86,184,370,663,537,867,783,823,287,61,131,268,506,197,306,133,272,512,834,592,153,311,575,762,445,229,112,226,441,756,294,335,616,460,173,350,635,792,1043,1071,1116,808,630,945,978,477,801,560,858,1078,948,1102,913,959,1037,1066,564,897,241,393,690,987,1108,947,944,883,730,423,216,22,49,109,221,430,585,581,912,676,380,191,90,189,379,674,742,278,273,513,641,631,734,307,442,757,683,386,99,93,193,384,680,980,612,499,263,128,59,10,11,25,56,122,248,479,804,1003,720,228,47,105,213,419,724,1008,1107,923,1044,670,544,140,285,532,861,397,695,611,313,580,728,928,320,591,922,1050,805,773,454,233,451,753,876,144,292,514,642,701,991,604,936,1104,769,1018,487,733,866,1082,999,934,300,558,890,621,838,515,836,839,1067,887,556,72,152,309,571,904,1084,977,672,377,187,88,7,19,43,98,203,402,462,237,116,52,23,8,20,45,102,208,411,518,840,809,488,256,124,253,484,711,205,394,198,392,689,986,831,332,609,554,165,31,70,32,71,151,305,567,627,952,1022,453,111,225,374,665,874,1089,1055,930,548,235,79,169,341,624,950,587,919,1096,953,617,675,246,476,473,697,989,1052,1100,1039,781,458,778,302,535,66,142,289,540,870,1035,1099,1101,909,329,605,255,425,731,1012,1114,750,772,946,619,336,390,195,388,685,983,1025,420,227,444,761,957,1015,736,1001,998,707,405,179,46,103,211,415,718,907,1098,965,649,359,177,180,361,651,855,1077,1020,845,1069,996,981,465,704,993,525,409,710,698,398,696,375,367,658,969,898,346,76,162,166,77,163,331,608,324,597,841,780,232,450,727,669,974,438,752,815,1014,1053,806,400,700,562,894,765,317,586,918,1061,355,646,716,938,164,334,614,941,1068,842,1040,784,832,508,270,28,63,136,277,522,849,1075,1046,794,469,242,118,239,464,789,1042,1095,884,550,295,145,67,29,64,138,281,526,856,543,878,1056,816,494,259,491,812,970,364,407,139,283,504,830,503,827,1065,1027,726,373,656,365,655,533,863,860,1080,1072,964,261,498,820,1059,873,280,524,854,598,293,114,34,75,160,326,600,170,343,563,347,171,344,120,245,158,322,594,457,777,1028,1011,837,510,149,301,493,735,976,882,846,1026,916,583,315,154,312,578,821,174,353,565,606,452,636,954,628,693,395,691,702,992,852,371,110,223,435,747,1021,1016,1110,1073,1112,1005,703,421,725,368,661,933,573,906,1097,967,653,362,372,664,207,410,713,921,927,1103,705,682,892,995,1109,1041,659,891,712,319,467,185,181,363,654,968,1010,668,973,971,660,714,214,418,722,1006,1113,1002,738,1017,895,746,434,137,279,523,853,787,480,200,396,345,629,351,637,937,899,640,352,638,413,577,73,155,316,584,917,1070,1094,1083,687,528,561,258,490,671,729,754,770,776,924,593,321,474,244,471,80,172,303,330,342,625,599,325,159,74,156,318,588,828,791,290,541,850,1024,741,497,819,847,931,1076,1081,862,559,182,366,266,502,825,1063,553,826,1064,829,530,101,206,408,709,966,648,126,260,495,817,1057,788,956,623,323,143,50,13,30,68,146,296,551,885,990,692,389,190,54,119,243,470,795,1047,1074,848,521,276,135,62,132,271,509,461,785,519,843,779,644,613,333,610,939,962,645,960,744,666,231,448,766,864,875,699,647,963,1090,871,994,751,997,376,422,439,449,767,1033,929,596,926,607,262,35,78,150,161,327,602,932,857,657,667,739,201,399,529,589,595,925,403,385,681,982,1086,520,745,872,798,472,240,84,83,178,360,650,915,1031,1000,717,414,210,44,100,204,406,708,719,427,737,639,958,760,443,437,749,814,486,249,95,196,167,337,620,811,768,1034,1013,732,382,209,412,496,818,1045,1058,1038,1087,869,539,288,141,286,536,865,236,459,782,910,943,1105,908,796,799,475,797,807,979,984,988,546,879,507,534,340,168,338,622,949,1092,1117,1088,706,186,215,224,436,748,1023,601,901,566,304,127,14,33,69,147,298,555,297,552,886,985,428,96,199,282,527,652,358,58,125,257,489,810,275,433,339,157,65,21,3,9,24,53,117,238,463,401,202,97,42,18,39,89,188,378,673,868,1085,903,570,308,148,299,557,888,517,568,274,516,456,775,889,328,603,935,1060,835,758,432,743,1019,1115,793,626,951,786,354,643,545,291,542,877,771,115,234,455,579,802,1051,576,590,920,1009,911,481,678,688,694,900,851,269,284,531,859,1079,902,1030,1091,1007,723,404,87,104,212,417,721,1004,803,478,247,121,55,4,12,27,60,129,264,500,822,961,679,219,41,94,194,387,684,416,218,426,348,632,538,134,222,431,492,254,92,40,91,192,381,677,314,447,740,429,220,108,48,106,217,424,549,618,774,1036,975,972,485,466,790,896,844,1054,940,582,914,881,1093,813,893,1049,800,1048,1106,955,764,1032,1029,1111,468,715,634,349,633,942,615,547,880,755,440,51,113,230,446,763,574,310,572,905,833,511,250,383,391,686,505,267,130,265,501,824,1062,569,759,662,369,183,85,26,57,123,252,483,356,175,81,36,15,5)Polycyclic presentation:
2,3,2,1117|a^2=c,d^a=d^903,d^b=d^996,d^c=d^1116105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13404, 8);