Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C24×C3 |
| direct product | C4×C9×C22⋊A4 |
| Fitting | C24×C4×C9 ⋊ C3 |
| derived | C24 ⋊ C3×C4×C9 |
| Sylow | C24×C4 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | C22⋊A4 |
| commutator, hypocenter | C24 | C3×C4×C9 |
| Frattini | C2×C3 | C2×C3×C22⋊A4 |
| Fitting | C24×C4×C9 | C3 |
| socle, cosocle | C25×C3 | C2×C32 |
| prime radical | C2×C3×C22⋊A4 | C2×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C4×C9 | |
870 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 870 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30929);