Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C22×D3×#121 |
| Fitting | C22×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ C23×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C22×C3×#121 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C22×#121 | D3 |
| hypocenter | C3 | C23×#121 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#121 | |
19998 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D3×D42, C22×D3×#447, C2×D3×#2847, … |
| maximal normal subgroup of | C22×D3×D42, C22×D3×#447, C2×D3×#2847, … |
| maximal quotient of | C22×D3×#392, C2×D3×#2847, C2×D3×#2410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076201);