Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C22×D3×C3⋊S4 |
| Fitting, socle | C24×C32 ⋊ C2×D3 |
| derived | C32×A4 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×C3⋊S4 |
| commutator, cosocle, hypocenter | C32×A4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | C2×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C22 | |
3663 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 3663 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×A4×C3⋊S4, C22×D32×S4, D3×D4×C3⋊S4, … |
| maximal normal subgroup of | D3×A4×C3⋊S4, C22×D32×S4, D3×D4×C3⋊S4, … |
| maximal quotient of | D3×D4×C3⋊S4, C22×D3×A4⋊S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47895);