Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D63 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D63 |
| Fitting | C2×C3×C7×C9 ⋊ C2 |
| derived | C7×C9 ⋊ C22×C3 |
| Hall | C3×C7×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ C2×D7 |
| Sylow | C7 ⋊ C2×C3×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D63 |
| commutator, cosocle, hypocenter | C7×C9 | C22×C3 |
| Frattini | C3 | C2×C3×D21 |
| Fitting | C2×C3×C7×C9 | C2 |
| socle | C2×C32×C7 | D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C2×C3 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
198 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D63⋊C4, C3×#92589, C3×#92592, … |
| maximal normal subgroup of | C3×D63⋊C4, C3×#92589, C3×#92592, … |
| maximal quotient of | C3×Dic126, C3×C4×D63, C3×D252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 125);