Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | ΓL(1,𝔽8)×#994 |
| Fitting, Hall | C7×H(5,𝔽2) ⋊ C32 |
| Sylow | H(5,𝔽2) ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#994 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C22×C32 |
| Fitting | C7×H(5,𝔽2) | C32 |
| socle | C2×C7 | C22×C3×A4 |
| hypercenter | Q8 | A4×ΓL(1,𝔽8) |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
228 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
95 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×Ω+(4,𝔽3), AGL(1,𝔽7)×#994, ΓL(1,𝔽8)×#5429, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×Ω+(4,𝔽3), AGL(1,𝔽7)×#994, ΓL(1,𝔽8)×#5429, … |
Past the SmallGroups library coverage — no SmallGroups id.