Finite Group DatabasebetaFinite Group Databasebeta| order | 31902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C409⋊C3)⋊D13 |
| Fitting, socle, derived, Hall | C13×C409 ⋊ C2×C3 |
| Hall | C13×C409⋊C3 ⋊ C2 |
| Hall | D5317 ⋊ C3 |
| Hall | C409⋊C3 ⋊ D13 |
| Sylow | C409 ⋊ C3×D13 |
| Sylow | C13 ⋊ D409⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C13×C409 | C2×C3 |
| LCS | G ▷ C13×C409 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 422 |
| transitive degree | μt(G) = 5317 |
A minimal faithful permutation representation — 2 generators on 422 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(15,19,20,16,17,21)(18,27,28,22,23,37)(24,40,41,30,31,54)(25,39,35,33,29,26)(32,59,60,43,44,77)(34,57,58,46,42,51)(36,38,56,48,49,52)(45,82,83,62,63,106)(47,80,81,65,61,74)(50,55,79,67,68,75)(53,95,93,71,72,70)(64,111,112,85,86,141)(66,109,110,88,84,103)(69,78,108,90,91,104)(73,124,125,96,97,99)(76,130,122,100,101,94)(87,146,147,114,115,182)(89,144,145,117,113,138)(92,107,143,119,120,139)(98,159,160,126,127,134)(102,166,164,131,132,129)(105,171,157,135,136,123)(116,187,188,149,150,229)(118,185,186,152,148,179)(121,142,184,154,155,180)(128,200,201,161,162,175)(133,207,208,167,168,170)(137,213,205,172,173,165)(140,218,198,176,177,158)(151,234,235,190,191,282)(153,232,233,193,189,226)(156,183,231,195,196,227)(163,247,248,202,203,222)(169,254,255,209,210,217)(174,261,259,214,215,212)(178,266,252,219,220,206)(181,271,245,223,224,199)(192,287,288,237,238,341)(194,285,286,240,236,279)(197,230,284,242,243,280)(204,300,301,249,250,275)(211,307,308,256,257,270)(216,314,315,262,263,265)(221,320,312,267,268,260)(225,325,305,272,273,253)(228,330,298,276,277,246)(239,346,347,290,291,396)(241,344,345,293,289,338)(244,283,343,295,296,339)(251,358,350,302,303,334)(258,364,365,309,310,329)(264,371,372,316,317,324)(269,377,375,321,322,319)(274,381,369,326,327,313)(278,386,362,331,332,306)(281,390,356,335,336,299)(292,366,367,349,311,410)(294,398,399,351,348,394)(297,342,397,353,354,395)(304,404,401,359,360,392)(318,414,355,373,374,385)(323,415,403,378,379,380)(328,416,409,382,383,376)(333,418,413,387,388,370)(337,419,408,389,391,363)(340,417,402,384,393,357)(352,411,412,400,368,421)(361,422,407,405,406,420);(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,15,18,24,32,45,64,87,116,151,192,239,292,350,286,188,231,271,266,261,254,247,232,187,230,281,337,387,331,276,196,115,148,162,168,173,177,180,182,226,275,329,385,353,295,242,195,154,119,90,67,48,35,20,19,25,34,47,66,89,118,153,194,241,294,352,401,345,235,284,330,325,320,314,307,300,285,234,283,340,382,326,272,223,155,86,113,127,132,136,139,141,179,222,270,324,380,402,356,298,245,198,157,122,93,58,28,39,27,38,53,73,98,128,163,204,251,304,361,407,399,288,343,390,386,381,377,371,364,358,344,287,342,378,321,267,219,176,120,63,84,97,101,104,106,138,175,217,265,319,376,408,362,305,252,205,164,125,81,41,56,57,40,55,76,102,133,169,211,258,311,368,406,412,347,397,417,419,418,416,415,414,410,404,398,346,373,316,262,214,172,135,91,44,61,72,75,77,103,134,170,212,260,313,370,413,369,312,259,208,160,110,60,79,95,80,59,78,105,137,174,216,264,318,291,348,360,367,374,379,383,388,391,393,395,396,421,422,411,366,309,256,209,167,131,100,68,31,42,52,54,74,99,129,165,206,253,306,363,409,375,315,255,201,145,83,108,130,124,109,82,107,140,178,221,269,323,354,238,289,303,310,317,322,327,332,336,339,341,394,420,405,359,302,249,202,161,126,96,71,49,23,29,37,51,70,94,123,158,199,246,299,357,403,372,308,248,186,112,143,171,166,159,144,111,142,181,225,274,328,384,296,191,236,250,257,263,268,273,277,280,282,338,392,400,351,293,240,193,152,117,88,65,46,33,17,21,26,36,50,69,92,121,156,197,244,297,355,365,301,233,147,184,218,213,207,200,185,146,183,228,278,333,389,335,243,150,189,203,210,215,220,224,227,229,279,334,349,290,237,190,149,114,85,62,43,30,22,16)Polycyclic presentation:
2,3,13,409|c^a=c^12,d^a=d^408,d^b=d^355908 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31902, 21);