Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | A4×C3⋊SL(2,ℤ/4) |
| unambiguous semidirect product | C3⋊(A4×SL(2,ℤ/4)) |
| unambiguous semidirect product | A4⋊(Dic3×A4) |
| derived | C22×C3×A4 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×C3⋊S4 |
| commutator, hypocenter | C22×C3×A4 | C3×C4 |
| Fitting, socle | C25×C3 | C3×D3 |
| prime radical | C2×C3×A42 | C2 |
| cosocle | C23×C3×A4 | C2×C3 |
| LCS | G ▷ C22×C3×A4 | |
| UCS | C1 ◁ C2 | |
470 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#64209, #72241499, A4×#64269, … |
| maximal normal subgroup of | A4×#64209, #72241499, A4×#64269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46389);