Finite Group DatabasebetaFinite Group Databasebeta| order | 21168 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | S4×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| Fitting, socle | C22×C72 ⋊ C2×C32×D3 |
| derived | C72×A4 ⋊ C22×C32 |
| Sylow | C72 ⋊ C2×C32×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C72×A4 | C22×C32 |
| Fitting, socle | C22×C72 | C2×C32×D3 |
| LCS | G ▷ C72×A4 | |
| UCS | C1 | |
1060 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
130 conjugacy classes — view the full character table.
| holomorph of | C22×ΓL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id.