Finite Group DatabasebetaFinite Group Databasebeta| order | 12501 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C463 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C463⋊C3 |
| commutator, socle, cosocle | C3×C463 | C32 |
| Fitting, prime radical | C32×C463 | C3 |
| hypercenter | C32 | C463⋊C3 |
| hypocenter | C463 | 3−1+2 |
| LCS | G ▷ C3×C463 ▷ C463 | |
| UCS | C1 ◁ C3 ◁ C32 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 472 |
| transitive degree | μt(G) = 4167 |
A minimal faithful permutation representation — 2 generators on 472 points:
(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,292,306,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,295,334,345,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,298,337,371,379,303,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,301,340,374,402,407,342,261,187,125,75,37,52,69,88,110,135,163,194,228,265,304,343,377,405,427,429,376,300,221,153,97,53,71,91,113,138,166,197,231,268,307,346,380,408,430,446,445,404,339,258,184,122,72,93,116,141,169,200,234,271,310,348,382,410,432,448,458,456,426,373,297,218,150,94,118,144,172,203,237,274,313,350,287,326,363,394,419,438,451,443,401,336,255,181,119,146,175,206,240,277,316,353,384,327,247,284,323,360,391,416,435,424,370,294,215,147,177,209,243,280,319,356,387,412,364,285,210,244,281,320,357,388,413,399,333,252,178,211,246,283,322,359,390,415,434,395,324,245,176,207,241,278,317,354,385,368,291,212,248,286,325,362,393,418,437,450,420,361,282,208,145,173,204,238,275,314,351,331,249,288,328,365,396,421,440,453,460,439,392,321,242,174,117,142,170,201,235,272,311,289,329,366,397,422,441,454,461,463,452,417,358,279,205,143,92,114,139,167,198,232,269,308,347,381,409,431,447,457,462,459,436,389,318,239,171,115,70,89,111,136,164,195,229,266,305,344,378,406,428,444,455,449,414,355,276,202,140,90,51,67,86,108,133,161,192,226,263,302,341,375,403,425,442,433,386,315,236,168,112,68,35,48,64,83,105,130,158,189,223,260,299,338,372,400,423,411,352,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,296,335,369,398,383,312,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,293,332,367,349,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,290,330,309,230,162,106,62,30,10)(465,468,469)(467,471,472);(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)(250,289,310)(253,291,326)(256,294,323)(259,297,320)(262,300,317)(265,303,314)(268,306,311)(271,309,308)(274,312,305)(277,315,302)(280,318,299)(283,321,296)(286,324,293)(288,327,290)(292,331,348)(295,333,363)(298,336,360)(301,339,357)(304,342,354)(307,345,351)(313,349,347)(316,352,344)(319,355,341)(322,358,338)(325,361,335)(328,364,332)(329,350,330)(334,368,382)(337,370,394)(340,373,391)(343,376,388)(346,379,385)(353,383,381)(356,386,378)(359,389,375)(362,392,372)(365,395,369)(366,384,367)(371,399,410)(374,401,419)(377,404,416)(380,407,413)(387,411,409)(390,414,406)(393,417,403)(396,420,400)(397,412,398)(402,424,432)(405,426,438)(408,429,435)(415,433,431)(418,436,428)(421,439,425)(422,434,423)(427,443,448)(430,445,451)(437,449,447)(440,452,444)(441,450,442)(446,456,458)(453,459,457)(454,460,455)(461,463,462)(464,465,467,466,469,471,470,468,472)Polycyclic presentation:
3^3,463|a^3=c,b^a=bc,d^a=d^211397 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #49154254, #49154255, #49154256, … |
| maximal normal subgroup of | #49154254, #49154255, #49154256, … |
| maximal quotient of | C32⋊(C463⋊C9), (C463⋊C9)⋊C9, C3×#27159752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12501, 12);