Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | ΓL(1,𝔽8)×#120⋊C3 |
| Fitting, Hall | C7×#120 ⋊ C32 |
| derived | C24×C7 ⋊ C2×C32 |
| Sylow | #120 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#120⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C7 | C2×C32 |
| Frattini | C22 | C2×A4×ΓL(1,𝔽8) |
| Fitting | C7×#120 | C32 |
| socle | C22×C7 | C2×C3×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 | |
172 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
50 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id.