Finite Group DatabasebetaFinite Group Databasebeta| order | 13428 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C373⋊C9) |
| unambiguous semidirect product | C373⋊(C22⋊C9) |
| derived, Hall | C22×C373 ⋊ C9 |
| Sylow | C373 ⋊ C22⋊C9 |
| Sylow | C22 ⋊ C373⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C22⋊(C373⋊C3) |
| commutator, hypocenter | C22×C373 | C9 |
| Fitting, socle, prime radical, cosocle | C22×C3×C373 | C3 |
| LCS | G ▷ C22×C373 | |
| UCS | C1 ◁ C3 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 386 |
| transitive degree | μt(G) = 6714 |
A minimal faithful permutation representation — 2 generators on 386 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)(179,212,246)(182,215,243)(185,218,240)(188,221,237)(191,213,234)(194,226,231)(197,229,228)(200,232,225)(203,235,223)(206,238,220)(209,241,217)(211,244,214)(216,249,279)(219,252,276)(222,255,273)(224,250,270)(227,260,267)(230,262,264)(233,265,261)(236,268,259)(239,271,257)(242,274,254)(245,277,251)(247,280,248)(253,283,307)(256,286,304)(258,284,301)(263,292,296)(266,294,293)(269,297,291)(272,299,290)(275,302,288)(278,305,285)(281,308,282)(287,311,329)(289,312,326)(295,318,319)(298,320,317)(300,322,316)(303,324,315)(306,327,313)(309,330,310)(314,333,345)(321,338,337)(323,339,336)(325,341,335)(328,343,334)(331,346,332)(340,352,351)(342,353,350)(344,355,349)(347,357,348)(354,362,361)(356,363,360)(358,365,359)(364,369,368)(366,370,367)(371,373,372)(375,376,377)(378,379,381,380,382,384,383,385,386);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,156,100,56,24,36,50,66,85,107,132,160,191,224,258,289,314,333,286,218,150,94,118,144,172,203,236,269,298,321,330,277,208,145,173,204,237,270,212,247,281,309,331,347,358,366,371,370,355,324,271,202,140,90,51,67,86,108,133,161,192,225,259,290,315,334,348,352,320,265,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,248,260,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,256,287,312,255,184,122,72,93,116,141,169,200,233,266,295,308,244,176,207,240,273,301,249,178,211,245,278,306,328,344,356,364,365,343,302,238,171,115,70,89,111,136,164,195,228,261,291,316,335,349,359,362,339,297,232,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,251,282,292,226,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,253,284,221,153,97,53,71,91,113,138,166,197,230,263,280,210,243,276,304,326,283,215,147,177,209,242,275,303,325,342,354,357,327,274,205,143,92,114,139,167,198,231,264,293,317,336,350,360,367,369,353,322,268,199,137,87,49,22,32,45,61,80,102,127,155,186,220,254,285,310,318,262,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,250,187,125,75,37,52,69,88,110,135,163,194,227,246,279,307,329,345,311,252,181,119,146,175,206,239,272,300,323,340,346,305,241,174,117,142,170,201,234,267,296,319,337,351,361,368,372,373,363,341,299,235,168,112,68,35,48,64,83,105,130,158,189,223,257,288,313,332,338,294,229,162,106,62,30,10)(374,377)(375,376)Polycyclic presentation:
3,2^2,373,3|a^3=e,b^a=c,c^a=bc,d^a=d^2841500 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13428, 15);