Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | #41×ΓL(1,𝔽8)⋊D7 |
| Fitting | C72×#41 ⋊ C2×C3 |
| Hall | #41×C7⋊D7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×#41 |
| Sylow | C72 ⋊ C2×C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×ΓL(1,𝔽8)⋊D7 |
| commutator, socle, cosocle | C2×C72 | C24×C3 |
| Frattini | C2 | C23×ΓL(1,𝔽8)⋊D7 |
| Fitting | C72×#41 | C2×C3 |
| hypercenter | #41 | ΓL(1,𝔽8)⋊D7 |
| hypocenter | C72 | C2×C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C4 ◁ #41 | |
820 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
220 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊D7, #41×ΓL(1,𝔽8)⋊AGL(1,𝔽7), H(5,𝔽2)×ΓL(1,𝔽8)⋊D7, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊D7, #41×ΓL(1,𝔽8)⋊AGL(1,𝔽7), H(5,𝔽2)×ΓL(1,𝔽8)⋊D7, … |
Past the SmallGroups library coverage — no SmallGroups id.