Finite Group DatabasebetaFinite Group Databasebeta| order | 28224 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×AGL(3,𝔽2) |
| Fitting, socle | C23×C7 ⋊ C3×GL(3,𝔽2) |
| derived | C7×AGL(3,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C7×AGL(3,𝔽2) | C3 |
| Fitting, socle, cosocle | C23×C7 | C3×GL(3,𝔽2) |
| LCS | G ▷ C7×AGL(3,𝔽2) | |
| UCS | C1 | |
426 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
55 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×AGL(3,𝔽2) |
| maximal normal subgroup of | AGL(1,𝔽7)×AGL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id.