Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)×D371 |
| unambiguous semidirect product | (C2×C53)⋊(D7×ΓL(1,𝔽8)) |
| Fitting, socle | C2×C72×C53 ⋊ C2×C3 |
| derived, Hall | C72×C53 ⋊ C22×C3 |
| Hall | C2×C7×D371 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8)×C53 ⋊ C22 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×D53 |
| Sylow | C53 ⋊ C2×D7×ΓL(1,𝔽8) |
| Sylow | C72 ⋊ C2×C3×D53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×D371 |
| commutator, cosocle, hypocenter | C72×C53 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72×C53 | C2×C3 |
| LCS | G ▷ C72×C53 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 4452 |
1870 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 122);