Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | D3×#2312 |
| Fitting | C3×#2312 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2312 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C22×C3 | C24×C4 |
| Fitting | C3×#2312 | C2 |
| socle, cosocle | C23×C3 | C25 |
| prime radical | C25×D3 | C22 |
| hypercenter | #2312 | D3 |
| hypocenter | C3 | C2×#2312 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2312 | |
4142 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 328145);