Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D3×#3585 |
| Fitting | C3×#3585 ⋊ C2 |
| Sylow | C3 ⋊ C2×#3585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D3×D4 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C3×#3585 | C2 |
| socle | C2×C3 | C24×D4 |
| hypercenter | #3585 | D3 |
| hypocenter | C3 | C2×#3585 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #3585 | |
11138 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
123 conjugacy classes — view the full character table.
| maximal subgroup of | #72213893, #72214349, D3×#23882, … |
| maximal normal subgroup of | D3×#23882, D3×#23872, D3×#61110, … |
| maximal quotient of | #72213893, #72214349, D3×#61451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083403);