Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | D3×#3573 |
| Fitting | C3×#3573 ⋊ C2 |
| Sylow | C3 ⋊ C2×#3573 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C24×D3 |
| commutator, socle | C2×C3 | C25×C4 |
| Frattini | C4 | C25×D3 |
| Fitting | C3×#3573 | C2 |
| prime radical | D3×#585 | C2 |
| cosocle | C3×C4 | C26 |
| hypercenter | #3573 | D3 |
| hypocenter | C3 | C2×#3573 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ #3573 | |
10858 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#60184, D3×#21478 |
| maximal normal subgroup of | D3×#60184, D3×#21478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083334);