Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×(C22⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×A4×(C22⋊C9)⋊C2 |
| derived | C22×C22⋊C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | C22×C22⋊C9 | C22×C3 |
| Frattini | C3 | C2×A4×S4 |
| Fitting, socle | C25×C3 | C3×D3 |
| LCS | G ▷ C22×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
638 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 638 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) = 54 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×((C22⋊C9)⋊C2)⋊C3, A4×#63694 |
| maximal normal subgroup of | C2×A4×((C22⋊C9)⋊C2)⋊C3, A4×#63694 |
| maximal quotient of | C2×C22⋊C9×(C22⋊C9)⋊C2, A4×#63694 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46102);