Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C7×D9 |
| Fitting | C2×C3×C7×C9 ⋊ C2 |
| derived | C9 ⋊ C22×C3×C7 |
| Hall | C3×C7×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | D9 |
| commutator, cosocle, hypocenter | C9 | C22×C3×C7 |
| Frattini | C3 | C2×C3×D3×C7 |
| Fitting | C2×C3×C7×C9 | C2 |
| socle | C2×C32×C7 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C3×C7 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D9×Dic7, C3×#92589, C3×#92591, … |
| maximal normal subgroup of | C3×D9×Dic7, C3×#92589, C3×#92591, … |
| maximal quotient of | C3×C7×Dic18, C3×C4×C7×D9, C3×C7×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 121);