Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#6516 |
| Fitting | C24×C72 ⋊ C2×C3 |
| Hall | C7×#6516 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ #120 |
| Sylow | C72 ⋊ C3×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C22×C72 | C23×C3 |
| Fitting | C24×C72 | C2×C3 |
| hypercenter | C24 | D7×ΓL(1,𝔽8) |
| hypocenter | C72 | C3×#120 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C24 | |
587 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 587 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) = 168 |
310 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72246111, #72246124, ΓL(1,𝔽8)×#89804 |
| maximal normal subgroup of | #72246111, #72246124, ΓL(1,𝔽8)×#89804 |
Past the SmallGroups library coverage — no SmallGroups id.