Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | Q8×#887 |
| Sylow | C3 ⋊ C4×Q82 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C22×C3 | C24×C4 |
| Fitting | C3×Q8×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C3 | C25 |
| hypercenter | Q8×C4⋊C4 | D3 |
| hypocenter | C3 | C4×Q82 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ Q8×C4⋊C4 | |
2324 conjugacy classes of subgroups.
| transitive degree | μt(G) = 768 |
150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 329570);