Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | A4×#387 |
| unambiguous semidirect product | (C22×#387)⋊C3 |
| Fitting, Sylow | C22×#387 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×A4 |
| commutator | C24 | C22×C3×C4 |
| Fitting | C22×#387 | C3 |
| socle, cosocle | C25 | C23×C3 |
| prime radical | C23×A4 | C23 |
| hypercenter | #387 | A4 |
| hypocenter | C22 | C3×#387 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C23 ◁ #387 | |
971 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084155);