Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×ΓL(1,𝔽8)×D53 |
| unambiguous semidirect product | C53⋊(C22×C7×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C53)⋊(C2×C7×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C7×D53)⋊(C3×C7) |
| Fitting, socle | C2×C72×C53 ⋊ C2×C3 |
| derived | C7×C53 ⋊ C22×C3×C7 |
| Hall | C2×C72×D53 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8)×C53 ⋊ C22 |
| Hall | C72×C53 ⋊ C22×C3 |
| Sylow | C53 ⋊ C22×C7×ΓL(1,𝔽8) |
| Sylow | C72 ⋊ C2×C3×D53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | ΓL(1,𝔽8)×D53 |
| commutator, cosocle, hypocenter | C7×C53 | C22×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72×C53 | C2×C3 |
| LCS | G ▷ C7×C53 | |
| UCS | C1 ◁ C2×C7 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2226 |
1960 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 116);