Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22 |
| direct product | S4×C32 |
| Fitting | C22×C32 ⋊ D3 |
| derived | A4 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | S4 |
| commutator, hypocenter | A4 | C2×C32 |
| Frattini | C16 | C2×S4 |
| Fitting | C22×C32 | D3 |
| socle | C23 | D3×C16 |
| prime radical | C2×S4 | C16 |
| cosocle | A4×C16 | C22 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C32 | |
125 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S4×C32, S4×M64, S4×D32, … |
| maximal normal subgroup of | C3×S4×C32, S4×M64, S4×D32, … |
| maximal quotient of | C32×GL(2,𝔽3), C3×S4×C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086039);