Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | S4×D16 |
| Fitting | C22×D16 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×S4 |
| commutator, cosocle | C8×A4 | C23 |
| Frattini | C8 | C22×S4 |
| Fitting | C22×D16 | D3 |
| socle | C23 | D3×D8 |
| hypercenter | D16 | S4 |
| hypocenter | A4 | C2×D16 |
| LCS | G ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D16 | |
440 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
55 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#509, S4×#508, C2×S4×D16, … |
| maximal normal subgroup of | S4×#509, S4×#508, C2×S4×D16, … |
| maximal quotient of | C2×S4×D16, S4×#357, S4×D32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088585);