Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | D42×Dic3 |
| Sylow | C3 ⋊ C4×D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C22×C3 | C24×C4 |
| Fitting, prime radical | C2×C3×D42 | C2 |
| socle, cosocle | C23×C3 | C25 |
| hypercenter | C2×D42 | D3 |
| hypocenter | C3 | C4×D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×D42 | |
4804 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
150 conjugacy classes — view the full character table.
| maximal quotient of | #72226124 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 329358);