Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C72×C13 · Schur multiplier (H2) = C7 |
| direct product | C13×#14558878 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C7×C13 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C72×C13 |
| prime radical | C13×H(3,𝔽7) | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7×C13 ◁ C72×C13 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 356 |
| transitive degree | μt(G) = 4459 |
A minimal faithful permutation representation — 2 generators on 356 points:
(1,2,5,13,31,68,114)(3,9,23,53,96,148,166)(4,8,16,34,71,117,167)(6,17,41,82,115,168,11)(7,20,49,95,147,205,165)(10,26,59,83,135,180,221)(12,22,40,75,122,116,171)(14,35,69,118,172,66,48)(15,38,80,133,192,113,21)(18,45,76,124,169,224,65)(19,44,85,125,170,223,29)(24,56,99,151,206,222,28)(25,43,32,72,123,179,220)(27,62,87,139,193,232,264)(30,52,39,81,70,121,178)(33,74,128,186,67,51,50)(36,78,119,176,231,263,61)(37,77,120,175,227,112,94)(42,86,134,173,228,163,93)(46,90,138,194,225,267,162)(47,54,97,149,207,226,111)(55,84,137,174,230,164,58)(57,79,131,177,233,266,110)(60,102,129,183,235,265,64)(63,107,152,187,241,279,296)(73,126,182,237,219,104,88)(89,130,181,229,270,262,146)(91,143,188,239,268,300,261)(92,100,127,184,240,269,218)(98,136,195,238,272,295,106)(101,140,185,242,274,299,217)(103,150,196,251,277,298,161)(105,141,189,236,273,297,109)(108,158,201,246,280,304,320)(132,190,234,275,302,294,145)(142,197,250,271,303,319,157)(144,202,252,291,301,325,318)(153,199,247,278,306,324,293)(154,208,245,281,310,323,260)(155,200,243,283,305,322,216)(156,191,248,276,308,321,160)(159,214,254,289,311,331,336)(198,244,282,307,327,335,204)(203,258,287,313,326,342,343)(209,255,285,315,330,341,334)(210,249,290,309,333,340,317)(211,256,288,312,329,339,292)(212,253,286,314,332,338,259)(213,257,284,316,328,337,215)(344,345,346,347,348,349,350,351,352,353,354,355,356);(1,3,10,27,63,108,159,4,11,28,64,109,160,215,12,29,65,110,161,216,259,30,66,111,162,217,260,292,67,112,163,218,261,293,317,113,164,219,262,294,318,334,165,220,263,295,319,335,343)(2,6,18,46,91,144,203,8,19,47,92,145,204,214,22,48,93,146,157,158,213,52,94,104,106,107,156,212,51,58,61,62,105,155,211,21,25,26,60,103,154,210,7,9,24,57,101,153,209)(5,14,36,79,132,191,249,16,37,59,90,142,200,255,40,55,56,100,152,208,258,39,43,45,89,141,199,254,50,23,54,98,150,202,257,15,17,42,87,140,198,253,20,44,88,102,143,201,256)(13,32,73,127,185,243,284,34,53,78,130,188,245,286,75,41,83,136,190,247,288,81,85,99,139,197,252,290,33,35,76,129,187,244,285,38,77,97,131,189,246,287,49,84,86,138,196,248,289)(31,69,119,177,234,276,309,71,120,135,194,250,283,315,122,137,151,184,241,281,313,70,72,124,181,236,278,311,74,96,149,195,251,291,316,80,82,134,193,242,282,314,95,125,126,183,239,280,312)(68,115,169,225,268,301,326,117,170,207,240,275,307,331,116,118,173,229,271,304,328,121,175,182,238,279,308,332,128,174,176,232,273,305,329,133,123,180,235,277,310,333,147,148,206,233,274,306,330)(114,166,221,264,296,320,336,167,168,222,265,297,321,337,171,223,224,266,298,322,338,178,172,226,267,299,323,339,186,227,228,269,300,324,340,192,230,237,270,302,325,341,205,179,231,272,303,327,342)1261 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 9);