Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×#547 |
| unambiguous semidirect product | (C22×#547)⋊C3 |
| Fitting, Sylow | C22×#547 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×A4 |
| commutator, socle, cosocle | C24 | C24×C3 |
| Fitting | C22×#547 | C3 |
| hypercenter | #547 | A4 |
| hypocenter | C22 | C3×#547 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ #547 | |
2525 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085050);