Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×D32 |
| unambiguous semidirect product | (C22×D32)⋊C3 |
| Fitting, Sylow | C22×D32 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×D16 |
| commutator, cosocle | C22×C16 | C22×C3 |
| Frattini | C16 | C22×A4 |
| Fitting | C22×D32 | C3 |
| socle | C23 | C3×D16 |
| hypercenter | D32 | A4 |
| hypocenter | C22 | C3×D32 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D32 | |
147 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#2263, S4×D32, #72342895 |
| maximal normal subgroup of | A4×#2263, S4×D32, #72342895 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084105);