Finite Group DatabasebetaFinite Group Databasebeta| order | 13251 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C7×C631 ⋊ C3 |
| Sylow | C631 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C631⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C631 | C3 |
| LCS | G ▷ C7×C631 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 638 |
| transitive degree | μt(G) = 4417 |
A minimal faithful permutation representation — 2 generators on 638 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,290)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(293,333,331)(296,336,368)(299,339,365)(302,342,362)(305,345,359)(308,348,356)(311,351,353)(314,354,350)(317,357,347)(320,360,344)(323,363,341)(326,366,338)(329,369,335)(330,334,332)(337,374,372)(340,377,406)(343,380,403)(346,383,400)(349,386,397)(352,389,394)(355,392,391)(358,395,388)(361,398,385)(364,401,382)(367,404,379)(370,407,376)(371,375,373)(378,412,410)(381,415,441)(384,418,438)(387,421,435)(390,424,432)(393,427,429)(396,430,426)(399,433,423)(402,436,420)(405,439,417)(408,442,414)(409,413,411)(416,447,445)(419,450,473)(422,453,470)(425,456,467)(428,459,464)(431,462,461)(434,465,458)(437,468,455)(440,471,452)(443,474,449)(444,448,446)(451,479,477)(454,482,502)(457,485,499)(460,488,496)(463,491,493)(466,494,490)(469,497,487)(472,500,484)(475,503,481)(476,480,478)(483,508,506)(486,511,528)(489,514,525)(492,517,522)(495,520,519)(498,523,516)(501,526,513)(504,529,510)(505,509,507)(512,534,532)(515,537,551)(518,540,548)(521,543,545)(524,546,542)(527,549,539)(530,552,536)(531,535,533)(538,557,555)(541,560,571)(544,563,568)(547,566,565)(550,569,562)(553,572,559)(554,558,556)(561,577,575)(564,580,588)(567,583,585)(570,586,582)(573,589,579)(574,578,576)(581,594,592)(584,597,602)(587,600,599)(590,603,596)(591,595,593)(598,608,606)(601,611,613)(604,614,610)(605,609,607)(612,619,617)(615,622,621)(616,620,618)(623,627,625)(624,628,626)(629,631,630)(633,634,635)(636,638,637);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,331,372,410,445,477,506,532,555,575,592,606,617,625,630,627,611,583,543,491,427,351,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,334,287,327,368,406,441,473,502,528,551,571,588,602,613,621,626,619,597,563,517,459,389,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,337,375,328,247,284,324,365,403,438,470,499,525,548,568,585,599,610,618,608,580,540,488,424,348,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,340,378,413,369,285,210,244,281,321,362,400,435,467,496,522,545,565,582,596,607,594,560,514,456,386,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,343,381,416,448,407,325,245,176,207,241,278,318,359,397,432,464,493,519,542,562,579,593,577,537,485,421,345,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,346,384,419,451,480,442,366,282,208,145,173,204,238,275,315,356,394,429,461,490,516,539,559,576,557,511,453,383,301,221,153,97,53,71,91,113,138,166,197,231,268,308,349,387,422,454,483,509,474,404,322,242,174,117,142,170,201,235,272,312,353,391,426,458,487,513,536,556,534,482,418,342,258,184,122,72,93,116,141,169,200,234,271,311,352,390,425,457,486,512,535,503,439,363,279,205,143,92,114,139,167,198,232,269,309,350,388,423,455,484,510,533,508,450,380,298,218,150,94,118,144,172,203,237,274,314,355,393,428,460,489,515,538,558,529,471,401,319,239,171,115,70,89,111,136,164,195,229,266,306,347,385,420,452,481,507,479,415,339,255,181,119,146,175,206,240,277,317,358,396,431,463,492,518,541,561,578,552,500,436,360,276,202,140,90,51,67,86,108,133,161,192,226,263,303,344,382,417,449,478,447,377,295,215,147,177,209,243,280,320,361,399,434,466,495,521,544,564,581,595,572,526,468,398,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,341,379,414,446,412,336,252,178,211,246,283,323,364,402,437,469,498,524,547,567,584,598,609,589,549,497,433,357,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,338,376,411,374,292,212,248,286,326,367,405,440,472,501,527,550,570,587,601,612,620,603,569,523,465,395,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,335,373,333,249,288,329,370,408,443,475,504,530,553,573,590,604,615,623,628,614,586,546,494,430,354,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,332,289,330,371,409,444,476,505,531,554,574,591,605,616,624,629,631,622,600,566,520,462,392,310,230,162,106,62,30,10)(632,633,634,636,635,637,638)Polycyclic presentation:
3,7,631|b^a=b^2,c^a=c^431475 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#27210397, #40997077, #40997081, … |
| maximal normal subgroup of | C2×#27210397, #40997077, #40997081, … |
| maximal quotient of | C2×#27210397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13251, 5);