Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#626 |
| Sylow | C32×C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D32 |
| commutator | C2×C32 | C22×C9 |
| Frattini | C2×C3 | C3×D32 |
| Fitting | C2×C32×C9 | C22 |
| socle, cosocle | C2×C33 | C22×C3 |
| prime radical | C3×#626 | C3 |
| hypocenter | C32 | D4×C9 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C9 | |
151 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#626, #13048299, #13048300, … |
| maximal normal subgroup of | D9×#626, #13048299, #13048300, … |
| maximal quotient of | C9×#3714, C9×#3716, C9×#3717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 365);