Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C73×C13 · Schur multiplier (H2) = C74 |
| direct product | C7×C13×H(3,𝔽7) |
| unambiguous semidirect product | C72⋊(C72×C13) |
| unambiguous semidirect product | (C72×C13)⋊C72 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C72×C13 | C72 |
| commutator, Frattini, cosocle | C7 | C73×C13 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C72×C13 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 4459 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5,6,7)(57,58,59,60,61,62,63,64,65,66,67,68,69);(8,9,11,15,22,33,42)(10,13,17,24,35,44,51)(12,18,28,40,49,56,50)(14,20,27,37,46,43,52)(16,25,38,48,55,41,19)(21,31,26,39,34,45,53)(23,36,47,54,32,30,29);(9,12,19,30,31,20,13)(11,16,26,17,18,29,27)(15,23,24,25,37,28,39)(22,34,40,46,38,35,36)(33,43,47,49,44,45,48)(42,51,52,53,54,55,56)5005 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 12);