Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #204 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C24×D24 |
| Fitting | C24×C3×C8 ⋊ C2 |
| Sylow | C3 ⋊ C24×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C25 | D12 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C24×C3×C8 | C2 |
| socle | C25×C3 | D4 |
| hypercenter | C24×C8 | D3 |
| hypocenter | C3 | C24×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C25 ◁ C24×C4 ◁ C24×C8 | |
15454 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083343);