Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#624 |
| derived | C32 ⋊ C2×C4×C9 |
| Sylow | C32×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D32 |
| commutator, hypocenter | C32 | C2×C4×C9 |
| Frattini | C2×C3 | C3×D32 |
| Fitting | C2×C32×C9 | C22 |
| socle, cosocle | C2×C33 | C22×C3 |
| prime radical | C2×C3×C3⋊D3 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C9 | |
143 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
162 conjugacy classes — view the full character table.
| maximal subgroup of | C9×D32⋊C4, C9×#3773, #13048246, … |
| maximal normal subgroup of | C9×D32⋊C4, C9×#3773, #13048246, … |
| maximal quotient of | C9×#3710, C9×#3712, C9×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 363);