Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C25×C3 |
| direct product | A4×C22⋊(C22⋊C9) |
| unambiguous semidirect product | (C22×A4)⋊(C22⋊C9) |
| derived, Sylow | C26 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A4×C22⋊A4 |
| commutator, hypocenter | C26 | C3×C9 |
| Fitting, socle, cosocle | C26×C3 | C32 |
| prime radical | C24×C3×A4 | C3 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C3 | |
1041 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | A4×(C22⋊A4)⋊C9, A4×#88881, A4×#88907, … |
| maximal normal subgroup of | A4×(C22⋊A4)⋊C9, A4×#88881, A4×#88907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46126);