Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×S4×C22⋊C9 |
| unambiguous semidirect product | A4⋊(C22×C22⋊C9) |
| unambiguous semidirect product | (C2×A4)⋊(C2×C22⋊C9) |
| unambiguous semidirect product | (C23×S4)⋊C9 |
| derived | C22×A4 ⋊ C22×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A4×S4 |
| commutator, hypocenter | C22×A4 | C22×C9 |
| Frattini | C3 | C2×A4×S4 |
| Fitting, socle | C25×C3 | C3×D3 |
| prime radical | C23×C3×S4 | C3 |
| cosocle | C22×C3×A4 | C22×C3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C2×C3 | |
650 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C2×S4×#1094 |
| maximal normal subgroup of | C2×S4×#1094 |
| maximal quotient of | C2×C22⋊C9×(C22⋊C9)⋊C2, C2×#72220166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46103);