Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×#532 |
| unambiguous semidirect product | (C22×#532)⋊C3 |
| Fitting, Sylow | C22×#532 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×A4 |
| commutator, socle, cosocle | C24 | C24×C3 |
| Frattini | C22 | C24×A4 |
| Fitting | C22×#532 | C3 |
| hypercenter | #532 | A4 |
| hypocenter | C22 | C3×#532 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
2663 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085004);