Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C13×C49 · Schur multiplier (H2) = C1 |
| direct product | C13×C343⋊C7 |
| unambiguous semidirect product | C343⋊(C7×C13) |
| unambiguous semidirect product | (C13×C343)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center | C13×C49 | C72 |
| commutator | C7 | C7×C13×C49 |
| Frattini, cosocle | C49 | C72×C13 |
| socle | C7×C13 | C7×C49 |
| prime radical | C72×C13 | C49 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C13×C49 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 356 |
| transitive degree | μt(G) = 4459 |
A minimal faithful permutation representation — 2 generators on 356 points:
(1,2,5,12,26,51,85,3,7,14,28,53,87,129,9,19,33,57,91,132,177,22,42,65,97,137,180,223,46,75,106,144,185,226,265,79,117,154,192,231,268,300,121,166,202,238,273,303,324,4,8,15,29,54,88,128,10,20,34,58,92,133,176,23,43,66,98,138,181,222,47,76,107,145,186,227,264,80,118,155,193,232,269,299,122,167,203,239,274,304,323,170,214,248,280,308,327,338,11,21,35,59,93,86,130,24,44,67,99,139,131,178,48,77,108,146,187,179,224,81,119,156,194,233,225,266,123,168,204,240,275,267,301,171,215,249,281,309,302,325,217,258,288,313,330,326,339,25,45,31,62,52,89,134,49,78,63,103,90,135,182,82,120,104,151,136,183,228,124,169,152,199,184,229,270,172,216,200,245,230,271,305,218,259,246,286,272,306,328,260,295,287,317,307,329,340,50,39,37,27,55,94,140,83,72,70,56,95,141,188,125,114,112,96,142,189,234,173,163,161,143,190,235,276,219,211,209,191,236,277,310,261,256,254,237,278,311,331,296,294,293,279,312,332,341,84,17,13,30,60,100,147,126,40,32,61,101,148,195,174,73,64,102,149,196,241,220,115,105,150,197,242,282,262,164,153,198,243,283,314,297,212,201,244,284,315,333,321,257,247,285,316,334,342,127,6,16,36,68,109,157,175,18,38,69,110,158,205,221,41,71,111,159,206,250,263,74,113,160,207,251,289,298,116,162,208,252,290,318,322,165,210,253,291,319,335,337,213,255,292,320,336,343)(344,345,346,347,348,349,350,351,352,353,354,355,356);(2,6,17,39,45,21,8)(5,13,31,15,16,37,35)(7,18,40,72,78,44,20)(12,27,29,30,59,36,62)(14,32,63,34,38,70,67)(19,41,73,114,120,77,43)(26,52,68,93,60,54,55)(28,56,58,61,99,69,103)(33,64,104,66,71,112,108)(42,74,115,163,169,119,76)(51,86,94,109,88,89,100)(53,90,110,139,101,92,95)(57,96,98,102,146,111,151)(65,105,152,107,113,161,156)(75,116,164,211,216,168,118)(85,128,130,134,140,147,157)(87,131,141,158,133,135,148)(91,136,159,187,149,138,142)(97,143,145,150,194,160,199)(106,153,200,155,162,209,204)(117,165,212,256,259,215,167)(129,176,178,182,188,195,205)(132,179,189,206,181,183,196)(137,184,207,233,197,186,190)(144,191,193,198,240,208,245)(154,201,246,203,210,254,249)(166,213,257,294,295,258,214)(177,222,224,228,234,241,250)(180,225,235,251,227,229,242)(185,230,252,275,243,232,236)(192,237,239,244,281,253,286)(202,247,287,248,255,293,288)(223,264,266,270,276,282,289)(226,267,277,290,269,271,283)(231,272,291,309,284,274,278)(238,279,280,285,313,292,317)(265,299,301,305,310,314,318)(268,302,311,319,304,306,315)(273,307,320,330,316,308,312)(300,323,325,328,331,333,335)(303,326,332,336,327,329,334)(324,338,339,340,341,342,343)5005 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 6);