Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×D3×Dic3 |
| unambiguous semidirect product | (C3×D3)⋊(C32×C4) |
| derived | C32 ⋊ C2×C32×C4 |
| Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D32 |
| commutator, hypocenter | C32 | C2×C32×C4 |
| Frattini | C2 | C32×D32 |
| Fitting, socle | C2×C34 | C22 |
| prime radical | C2×C33×D3 | C2 |
| cosocle | C2×C32 | C22×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C32 | |
394 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
162 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D32×Dic3, C3×D3×#624, C3×D3×#626, … |
| maximal normal subgroup of | C3×D32×Dic3, C3×D3×#624, C3×D3×#626, … |
| maximal quotient of | C32×D3×C3⋊C8, C32×#3711, C32×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 646);