Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D12×C22⋊Q8 |
| Fitting | C3×C4×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ D4×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C3×C4×C22⋊Q8 | C2 |
| prime radical | C23×D12 | C22 |
| hypercenter | C4×C22⋊Q8 | D3 |
| hypocenter | C3 | D4×C22⋊Q8 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 | |
3488 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 364895);