Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×ΓL(1,𝔽8)⋊#6157 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C7×#6157) |
| unambiguous semidirect product | (C7⋊#6157)⋊(C3×C7) |
| Fitting, socle, Hall | C72×C53 ⋊ C3×C4 |
| derived | C7×C53 ⋊ C3×C4×C7 |
| Hall | C7×C7⋊#6157 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8)×C53 ⋊ C4 |
| Hall | C7×ΓL(1,𝔽8) ⋊ #6157 |
| Sylow | C53 ⋊ C7×ΓL(1,𝔽8)⋊C4 |
| Sylow | C72 ⋊ C3×#6157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | ΓL(1,𝔽8)⋊#6157 |
| commutator, hypocenter | C7×C53 | C3×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C72×C53 | C3×C4 |
| prime radical | C7×ΓL(1,𝔽8)×D53 | C2 |
| cosocle | C7×D53 | C2×C3×C7 |
| LCS | G ▷ C7×C53 | |
| UCS | C1 ◁ C7 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2226 |
553 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 88);