Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C2×#87571 |
| Fitting | C22×C3×2−1+4 ⋊ C2 |
| Sylow | C3 ⋊ C2×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C25×D3 |
| commutator, cosocle | C2×C3 | C27 |
| Frattini | C2 | C26×D3 |
| Fitting | C22×C3×2−1+4 | C2 |
| socle | C22×C3 | C26 |
| hypercenter | C22×2−1+4 | D3 |
| hypocenter | C3 | C2×2−1+6 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×2−1+4 | |
59186 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083467);