Finite Group DatabasebetaFinite Group Databasebeta| order | 12747 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C7×C607 ⋊ C3 |
| Sylow | C607 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C607⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C607 | C3 |
| LCS | G ▷ C7×C607 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 614 |
| transitive degree | μt(G) = 4249 |
A minimal faithful permutation representation — 2 generators on 614 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)(290,332,376)(293,335,373)(296,338,370)(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)(326,368,340)(329,371,337)(331,374,334)(333,378,399)(336,380,421)(339,383,418)(342,386,415)(345,389,412)(348,392,409)(351,395,406)(354,398,403)(357,401,400)(360,404,397)(363,407,394)(366,410,391)(369,413,388)(372,416,385)(375,419,382)(377,422,379)(381,426,443)(384,428,464)(387,431,461)(390,434,458)(393,437,455)(396,440,452)(402,445,447)(405,448,444)(408,450,442)(411,453,439)(414,456,436)(417,459,433)(420,462,430)(423,465,427)(424,446,425)(429,469,483)(432,471,501)(435,474,498)(438,477,495)(441,480,492)(449,486,485)(451,488,484)(454,490,482)(457,493,479)(460,496,476)(463,499,473)(466,502,470)(467,487,468)(472,506,517)(475,508,532)(478,511,529)(481,514,526)(489,520,519)(491,522,518)(494,524,516)(497,527,513)(500,530,510)(503,533,507)(504,521,505)(509,537,545)(512,539,557)(515,542,554)(523,548,547)(525,550,546)(528,552,544)(531,555,541)(534,558,538)(535,549,536)(540,562,567)(543,564,576)(551,570,569)(553,572,568)(556,574,566)(559,577,563)(560,571,561)(565,581,583)(573,586,585)(575,588,584)(578,590,582)(579,587,580)(589,596,595)(591,598,594)(592,597,593)(599,603,602)(600,604,601)(605,607,606)(609,610,611)(612,614,613);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,387,432,472,480,392,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,396,441,481,515,543,565,564,511,434,341,255,181,119,146,175,206,240,277,317,360,405,449,487,422,330,373,418,461,498,529,554,537,471,383,292,212,248,286,326,369,414,457,494,525,551,571,533,462,371,285,210,244,281,321,364,409,452,426,332,377,423,466,503,534,559,578,591,599,604,590,555,496,413,322,242,174,117,142,170,201,235,272,312,355,400,444,484,518,546,568,584,594,601,596,572,524,453,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,391,436,476,510,538,561,548,488,404,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,382,427,468,445,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,381,395,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,390,435,475,509,514,437,344,258,184,122,72,93,116,141,169,200,234,271,311,354,399,443,483,517,545,567,583,581,539,474,386,295,215,147,177,209,243,280,320,363,408,451,489,521,465,374,287,327,370,415,458,495,526,506,428,335,249,288,329,372,417,460,497,528,553,573,587,558,499,416,325,245,176,207,241,278,318,361,406,378,424,467,504,535,560,579,592,600,605,607,598,574,527,456,365,279,205,143,92,114,139,167,198,232,269,309,352,397,442,482,516,544,566,582,593,586,550,490,407,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,388,433,473,507,536,520,448,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,379,425,398,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,384,429,440,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,393,438,478,512,540,542,477,389,298,218,150,94,118,144,172,203,237,274,314,357,402,446,376,421,464,501,532,557,576,562,508,431,338,252,178,211,246,283,323,366,411,454,491,523,549,502,419,328,247,284,324,367,412,455,492,469,380,289,331,375,420,463,500,531,556,575,589,597,577,530,459,368,282,208,145,173,204,238,275,315,358,403,447,485,519,547,569,585,595,602,606,603,588,552,493,410,319,239,171,115,70,89,111,136,164,195,229,266,306,349,394,439,479,513,541,563,580,570,522,450,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,385,430,470,505,486,401,310,230,162,106,62,30,10)(608,609,610,612,611,613,614)Polycyclic presentation:
3,7,607|b^a=b^2,c^a=c^2101419 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#27164995, #38783787, #38783791, … |
| maximal normal subgroup of | C2×#27164995, #38783787, #38783791, … |
| maximal quotient of | C2×#27164995, #50304689, C3×#27164995 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12747, 3);