Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D32×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | D3×A4×C3⋊C8 |
| unambiguous semidirect product | (C3×D3)⋊(C8×A4) |
| unambiguous semidirect product | (C22×D3×C3⋊C8)⋊C3 |
| derived | C22×C32 ⋊ C2×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32×A4 |
| commutator, hypocenter | C22×C32 | C2×C3×C8 |
| Fitting, cosocle | C22×C32×C4 | C22×C3 |
| socle | C23×C32 | C2×C3×C4 |
| prime radical | C2×C3×D3×A4 | C4 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C4 | |
458 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32187);