Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C8 |
| direct product | C2×#68062 |
| Fitting | C2×C3×C82 ⋊ C2 |
| Sylow | C3 ⋊ C2×#1712 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | #874 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C2×C3×C82 | C2 |
| socle | C23×C3 | #127 |
| hypercenter | C2×C82 | D3 |
| hypocenter | C3 | C2×#1712 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 | |
1342 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | #72225817, #72225871, #72225879, … |
| maximal normal subgroup of | #72225817, #72225871, #72225879, … |
| maximal quotient of | #72225817 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 157259);