Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C22×#5966 |
| Fitting | C22×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ C22×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C24×D3 |
| commutator, cosocle | C2×C3 | C27 |
| Frattini | C2 | C26×D3 |
| Fitting | C22×C3×H(5,𝔽2) | C2 |
| socle | C23×C3 | C25 |
| hypercenter | C22×H(5,𝔽2) | D3 |
| hypocenter | C3 | C22×#585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×H(5,𝔽2) | |
60366 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#87204, C22×D3×#585 |
| maximal normal subgroup of | C22×#87204, C22×D3×#585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083460);