Finite Group DatabasebetaFinite Group Databasebeta| order | 11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)×D137 |
| unambiguous semidirect product | C137⋊(C22×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C137)⋊(C2×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C7×D137)⋊C3 |
| Fitting, socle | C2×C7×C137 ⋊ C2×C3 |
| derived, Hall | C7×C137 ⋊ C22×C3 |
| Hall | C2×C7×D137 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C137 ⋊ C22 |
| Sylow | C137 ⋊ C22×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×C3×D137 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×D137 |
| commutator, cosocle, hypocenter | C7×C137 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C137 | C2×C3 |
| LCS | G ▷ C7×C137 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
700 conjugacy classes — view the full character table.
| maximal subgroup of | D3×ΓL(1,𝔽8)×D137, C2×C3×ΓL(1,𝔽8)×D137, C2×ΓL(1,𝔽8)×D411 |
| maximal normal subgroup of | C2×C3×ΓL(1,𝔽8)×D137 |
| maximal quotient of | C2×C7⋊C9×D137, D3×ΓL(1,𝔽8)×D137, C2×C3×ΓL(1,𝔽8)×D137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(11508, 19);