Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×A4×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | D3×A4×S4 |
| unambiguous semidirect product | (C3×S4)⋊(C2×A4) |
| unambiguous semidirect product | (D3×A4)⋊(C2×A4) |
| unambiguous semidirect product | (C22×D3×S4)⋊C3 |
| Fitting, socle | C24×C3 ⋊ C2×C3×D3 |
| derived | C22×C3×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C3×A4 | C22×C3 |
| Fitting, socle | C24×C3 | C2×C3×D3 |
| LCS | G ▷ C22×C3×A4 | |
| UCS | C1 | |
765 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 765 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | D3×S42, A4×#89101, S4×(C3⋊D3)⋊A4, … |
| maximal normal subgroup of | D3×S42, C3×D3×A4×S4, C2×D3×A4×S4 |
| maximal quotient of | A4×#89101, S4×(C3⋊D3)⋊A4, A4×((C3⋊S4)⋊C3)⋊C2, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47857);