Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D32×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×D32×A4 |
| unambiguous semidirect product | (C3×D3)⋊(C23×A4) |
| Fitting, socle | C24×C32 ⋊ C22×C3 |
| derived | C22×C32 ⋊ C24×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D32×A4 |
| commutator, cosocle, hypocenter | C22×C32 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | C22×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C22 | |
4986 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 4986 conjugacy classes of subgroups — too many to draw. Only its 312 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | D32×A42, D32×GL(2,ℤ/4), C22×D32×S4, … |
| maximal normal subgroup of | D32×A42, D32×GL(2,ℤ/4), C22×D32×S4, … |
| maximal quotient of | D32×D4×A4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47901);