Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C9⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C4×C9⋊S4 |
| Fitting | C23×C4×C9 ⋊ D3 |
| derived | C9×A4 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C9⋊S4 |
| commutator, hypocenter | C9×A4 | C22×C4 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C23×C4×C9 | D3 |
| socle | C24×C3 | C2×C3⋊D3 |
| prime radical | C22×C9⋊S4 | C2 |
| cosocle | C2×C9×A4 | C23 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 ◁ C2×C4 | |
915 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 915 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30513);