Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D32×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D32×D9 |
| unambiguous semidirect product | (C3×D3×D9)⋊C2 |
| unambiguous semidirect product | (D9×C3⋊D3)⋊C2 |
| unambiguous semidirect product | (D3×C3⋊D9)⋊C2 |
| Fitting, derived, Sylow | C32×C9 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C9 | C23 |
| Frattini | C3 | D33 |
| socle | C33 | C22×D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 | |
255 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | D9×GO+(4,𝔽2), C2×D32×D9, D3×D92, … |
| maximal normal subgroup of | D9×GO+(4,𝔽2), C2×D32×D9, C3×D32×D9, … |
| maximal quotient of | D3×#6189, #13048283, D3×Dic3×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 554);