Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D4×D7×ΓL(1,𝔽8) |
| Fitting | C72×D4 ⋊ C2×C3 |
| Hall | C7×D4×D7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×D4 |
| Sylow | C72 ⋊ C2×C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C72×D4 | C2×C3 |
| hypercenter | D4 | D7×ΓL(1,𝔽8) |
| hypocenter | C72 | C2×C3×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ D4 | |
224 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
125 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#6548, #72245804, D4×D7×AGL(1,𝔽7), … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#6548, #72245804, D4×D7×AGL(1,𝔽7), … |
| maximal quotient of | ΓL(1,𝔽8)×#6500 |
Past the SmallGroups library coverage — no SmallGroups id.