Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #564⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×#564⋊C9 |
| unambiguous semidirect product | (C3×#564)⋊C9 |
| derived, Sylow | #564 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #564⋊C3 |
| commutator, hypocenter | #564 | C3×C9 |
| Frattini | C22×C3 | C3×C22⋊A4 |
| Fitting | C32×#564 | C3 |
| socle, prime radical | C22×C32 | C22⋊A4 |
| cosocle | C3×#564 | C32 |
| LCS | G ▷ #564 | |
| UCS | C1 ◁ C32 | |
236 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18331);