Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×D32 |
| unambiguous semidirect product | (C3×D3)⋊(C22×C32) |
| Fitting, socle | C2×C34 ⋊ C22 |
| derived | C32 ⋊ C23×C32 |
| Sylow | C34 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D32 |
| commutator, cosocle, hypocenter | C32 | C23×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C32 | |
630 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 168 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 | C32×D32⋊C4, C3×D32⋊Dic3, D32⋊(C3⋊Dic3), … |
| maximal normal subgroup of | C32×D32⋊C4, C3×D32⋊Dic3, D32⋊(C3⋊Dic3), … |
| maximal quotient of | C32×D3×Dic6, C32×#3795, C32×#3796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 746);