Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#5644 |
| Fitting | C22×C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ C22×#538 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C22×C3×C22⋊Q8 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C22×C22⋊Q8 | D3 |
| hypocenter | C3 | C22×#538 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C22⋊Q8 | |
9566 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076300);