Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×A4×C3⋊S4 |
| unambiguous semidirect product | (C2×C3)⋊(A4×S4) |
| unambiguous semidirect product | (C2×A4)⋊(D3×A4) |
| Fitting, socle | C25×C3 ⋊ C3×D3 |
| derived | C22×C3×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×C3⋊S4 |
| commutator, cosocle, hypocenter | C22×C3×A4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | C3×D3 |
| LCS | G ▷ C22×C3×A4 | |
| UCS | C1 ◁ C2 | |
926 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 926 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×A4×S4, D3×A4×C3⋊S4, C2×S4×C3⋊S4, … |
| maximal normal subgroup of | C2×D3×A4×S4, C2×S4×C3⋊S4, C2×#13181790, … |
| maximal quotient of | D3×A4×C3⋊S4, A4×#64269, #72242417 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47876);