Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C72×C13 · Schur multiplier (H2) = C72 |
| direct product | C13×H(3,𝔽7)⋊C7 |
| unambiguous semidirect product | H(3,𝔽7)⋊(C7×C13) |
| unambiguous semidirect product | (C13×H(3,𝔽7))⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C7×C13 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C72×C13 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7×C13 ◁ C72×C13 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 637 |
A minimal faithful permutation representation — 2 generators on 62 points:
(1,2,4,8,16,27,35)(3,5,11,19,29,36,44)(6,13,17,30,40,47,43)(7,12,23,32,39,37,45)(9,20,28,38,46,15,24)(10,22,33,42,49,34,14)(18,31,41,48,26,25,21)(50,51,52,53,54,55,56,57,58,59,60,61,62);(2,5,12,24,25,14,6)(4,9,13,23,10,11,21)(8,17,22,18,20,32,19)(16,28,30,39,33,29,31)(27,36,37,38,41,42,40)1261 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 7);