Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | ΓL(1,𝔽8)×#6548 |
| Fitting | C2×C72×D4 ⋊ C2×C3 |
| Hall | C7×#6548 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ H(5,𝔽2) |
| Sylow | C72 ⋊ C3×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D7×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C72 | C24×C3 |
| Fitting | C2×C72×D4 | C2×C3 |
| hypercenter | C2×D4 | D7×ΓL(1,𝔽8) |
| hypocenter | C72 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
691 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
235 conjugacy classes — view the full character table.
| maximal subgroup of | #72245862, ΓL(1,𝔽8)×#89849, #72245941, … |
| maximal normal subgroup of | #72245862, ΓL(1,𝔽8)×#89849, #72245941, … |
Past the SmallGroups library coverage — no SmallGroups id.