Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| direct product | S4×GL(3,𝔽2) |
| Fitting | C22 ⋊ D3×GL(3,𝔽2) |
| socle | C22×GL(3,𝔽2) ⋊ D3 |
| derived | A4×GL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | A4×GL(3,𝔽2) | C2 |
| Fitting | C22 | D3×GL(3,𝔽2) |
| socle | C22×GL(3,𝔽2) | D3 |
| cosocle | A4 | C2×GL(3,𝔽2) |
| LCS | G ▷ A4×GL(3,𝔽2) | |
| UCS | C1 | |
312 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
30 conjugacy classes — view the full character table.
| maximal subgroup of | GL(3,𝔽2)2, GL(3,𝔽2)×A6, S4×AGL(3,𝔽2) |
| maximal quotient of | GL(2,𝔽3)×GL(3,𝔽2), #72221415, S4×SL(2,𝔽7), … |
| automorphism group of | C23×A4 |
Past the SmallGroups library coverage — no SmallGroups id.