Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | S4×M32 |
| Fitting | C22×M32 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×S4 |
| commutator | C2×A4 | C22×C8 |
| Fitting | C22×M32 | D3 |
| socle | C23 | C2×D3×C8 |
| prime radical | C22×S4 | C8 |
| cosocle | C8×A4 | C23 |
| hypercenter | M32 | S4 |
| hypocenter | A4 | C2×M32 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C8 ◁ M32 | |
399 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#509, S4×#360, S4×#365, … |
| maximal normal subgroup of | S4×#509, S4×#360, S4×#365, … |
| maximal quotient of | S4×#346 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088515);