Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | AGL(1,𝔽7)×ΓL(1,𝔽8)⋊C4 |
| derived, Sylow | C72 ⋊ C2×C32×C4 |
| Hall | ΓL(1,𝔽8)2 ⋊ C2×C4 |
| Hall | D7×Dic7 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽7)2 |
| commutator, hypocenter | C72 | C2×C32×C4 |
| Fitting, socle, cosocle | C2×C72 | C22×C32 |
| prime radical | C2×ΓL(1,𝔽8)×AGL(1,𝔽7) | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C2 | |
198 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
98 conjugacy classes — view the full character table.
| maximal subgroup of | #72245286, #72245296, C4×AGL(1,𝔽7)2, … |
| maximal normal subgroup of | #72245286, #72245296, C4×AGL(1,𝔽7)2 |
| maximal quotient of | AGL(1,𝔽7)×ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id.