Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D32⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D32⋊D3 |
| unambiguous semidirect product | D32⋊(C3×D3) |
| Fitting, socle, Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D32⋊D3 |
| commutator, cosocle | C3×C3⋊D3 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C34 | D4 |
| hypocenter | C33 | C3×D4 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C3 | |
187 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 12 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×GO+(4,𝔽2), D3×D32⋊D3, (D32⋊D3)⋊D3, … |
| maximal normal subgroup of | C3×D3×GO+(4,𝔽2), D3×D32⋊D3, (D32⋊D3)⋊D3, … |
| maximal quotient of | C3×#88932, C3×D32⋊Dic3, C3×#88934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 719);