Finite Group DatabasebetaFinite Group Databasebeta| order | 12873 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C7×C613 ⋊ C3 |
| Sylow | C613 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C613⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C613 | C3 |
| LCS | G ▷ C7×C613 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 620 |
| transitive degree | μt(G) = 4291 |
A minimal faithful permutation representation — 2 generators on 620 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)(250,289,330)(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)(290,332,375)(293,335,372)(296,338,369)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,333,340)(329,370,337)(331,373,334)(336,378,417)(339,381,414)(342,384,412)(345,387,410)(348,390,407)(351,393,404)(354,396,401)(357,399,398)(360,402,395)(363,405,392)(366,408,389)(368,379,386)(371,413,383)(374,415,380)(376,418,377)(382,421,454)(385,424,452)(388,426,451)(391,429,449)(394,432,446)(397,435,443)(400,438,440)(403,441,437)(406,444,434)(409,447,431)(411,422,428)(416,453,423)(419,455,420)(425,458,486)(427,460,485)(430,462,484)(433,465,482)(436,468,479)(439,471,476)(442,474,473)(445,477,470)(448,480,467)(450,459,464)(456,487,457)(461,488,513)(463,490,512)(466,492,511)(469,495,509)(472,498,506)(475,501,503)(478,504,500)(481,507,497)(483,489,494)(491,514,536)(493,516,535)(496,518,534)(499,521,532)(502,524,529)(505,527,526)(508,530,523)(510,515,520)(517,537,556)(519,539,555)(522,541,554)(525,544,552)(528,547,549)(531,550,546)(533,538,543)(540,557,573)(542,559,572)(545,561,571)(548,564,569)(551,567,566)(553,558,563)(560,574,587)(562,576,586)(565,578,585)(568,581,583)(570,575,580)(577,588,598)(579,590,597)(582,592,596)(584,589,594)(591,599,606)(593,601,605)(595,600,604)(602,607,611)(603,608,610)(609,612,613)(615,616,617)(618,620,619);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,282,208,145,173,204,238,275,315,358,401,440,473,500,523,543,514,460,384,295,215,147,177,209,243,280,320,363,406,445,478,505,528,548,565,579,591,600,581,547,501,438,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,379,322,242,174,117,142,170,201,235,272,312,355,398,437,470,497,520,488,424,338,252,178,211,246,283,323,366,409,448,481,508,531,551,568,582,593,602,608,592,564,524,471,396,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,382,422,365,279,205,143,92,114,139,167,198,232,269,309,352,395,434,467,494,458,381,292,212,248,286,326,368,411,450,483,510,533,553,570,584,595,603,609,612,601,578,544,498,435,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,385,425,459,408,319,239,171,115,70,89,111,136,164,195,229,266,306,349,392,431,464,421,335,249,288,329,371,375,417,454,486,513,536,556,573,587,598,606,611,613,607,590,561,521,468,393,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,388,427,461,489,447,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,389,428,378,289,331,374,416,418,330,372,414,452,485,512,535,555,572,586,597,605,610,599,576,541,495,432,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,391,430,463,491,515,480,405,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,386,332,376,419,456,455,373,287,327,369,412,451,484,511,534,554,571,585,596,604,588,559,518,465,390,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,394,433,466,493,517,538,507,444,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,383,423,457,487,415,328,247,284,324,367,410,449,482,509,532,552,569,583,594,574,539,492,429,344,258,184,122,72,93,116,141,169,200,234,271,311,354,397,436,469,496,519,540,558,530,477,402,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,380,420,453,370,285,210,244,281,321,364,407,446,479,506,529,549,566,580,557,516,462,387,298,218,150,94,118,144,172,203,237,274,314,357,400,439,472,499,522,542,560,575,550,504,441,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,377,413,325,245,176,207,241,278,318,361,404,443,476,503,526,546,563,537,490,426,341,255,181,119,146,175,206,240,277,317,360,403,442,475,502,525,545,562,577,589,567,527,474,399,310,230,162,106,62,30,10)(614,615,617,619,616,618,620)Polycyclic presentation:
3,7,613|b^a=b^4,c^a=c^651433 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#27166121, #38788405, #38788409, … |
| maximal normal subgroup of | C2×#27166121, #38788405, #38788409, … |
| maximal quotient of | C2×#27166121, #50311065, C3×#27166121 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12873, 4);