Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C29 |
| direct product | ΓL(1,𝔽8)×#2846 |
| Fitting, Hall | C7×#2846 ⋊ C3 |
| Sylow | C7 ⋊ C3×#2846 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C23×C7 | C24×C3 |
| Fitting | C7×#2846 | C3 |
| hypercenter | #2846 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#2846 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #2846 | |
6844 conjugacy classes of subgroups; only the 540 normal subgroups are shown.
This group has 6844 conjugacy classes of subgroups — too many to draw. Only its 540 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
220 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#2846, ΓL(1,𝔽8)×#87524, ΓL(1,𝔽8)×#23900, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#2846, ΓL(1,𝔽8)×#23900, #72276924, … |
| maximal quotient of | ΓL(1,𝔽8)×#32898 |
Past the SmallGroups library coverage — no SmallGroups id.