Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | ΓL(1,𝔽8)×#874 |
| Fitting | C3×C42×C7 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #127 |
| Sylow | C7 ⋊ C3×#874 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C22×C3×C7 | C23×C3 |
| Fitting | C3×C42×C7 | C2×C3 |
| hypercenter | C42 | D3×ΓL(1,𝔽8) |
| hypocenter | C3×C7 | C3×#127 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | D3×ΓL(1,𝔽8)×#127, AGL(1,𝔽7)×#874, #72244271, … |
| maximal normal subgroup of | D3×ΓL(1,𝔽8)×#127, AGL(1,𝔽7)×#874, #72244271, … |
| maximal quotient of | ΓL(1,𝔽8)×#64083 |
Past the SmallGroups library coverage — no SmallGroups id.