Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C8 |
| derived, Sylow | C72 ⋊ C32×C8 |
| Hall | ΓL(1,𝔽8)2 ⋊ C8 |
| Hall | C7×C7⋊C8 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | ΓL(1,𝔽8)×AGL(1,𝔽7) |
| commutator, hypocenter | C72 | C32×C8 |
| Fitting, cosocle | C4×C72 | C2×C32 |
| socle | C2×C72 | C32×C4 |
| prime radical | C2×ΓL(1,𝔽8)2 | C4 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C4 | |
102 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 102 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #72245135, ΓL(1,𝔽8)×#1136⋊C3, #72245145, … |
| maximal normal subgroup of | #72245135, ΓL(1,𝔽8)×#1136⋊C3, #72245145, … |
Past the SmallGroups library coverage — no SmallGroups id.