Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×D3×#538 |
| Fitting | C2×C3×#538 ⋊ C2 |
| Sylow | C3 ⋊ C22×#538 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C24×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C2×C3×#538 | C2 |
| socle | C23×C3 | C25 |
| hypercenter | C2×#538 | D3 |
| hypocenter | C3 | C22×#538 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#538 | |
9566 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#37660, D3×#62863, D3×#44228, … |
| maximal normal subgroup of | D3×#37660, D3×#62863, D3×#44228, … |
| maximal quotient of | C2×D3×#2668, D3×#37660, C2×D3×#2631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076291);