Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×#577 |
| Fitting, Sylow | C22×#577 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D4×A4 |
| commutator, cosocle | C22×C4 | C24×C3 |
| Frattini | C4 | C24×A4 |
| Fitting | C22×#577 | C3 |
| socle | C23 | C22×C3×D4 |
| hypercenter | #577 | A4 |
| hypocenter | C22 | C3×#577 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #577 | |
2283 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085268);