Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×#338⋊C9 |
| unambiguous semidirect product | (C3×#338)⋊C9 |
| derived, Sylow | #338 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C4 | C42⋊C3 |
| commutator, hypocenter | #338 | C3×C9 |
| Frattini | C22×C3×C4 | C3×A4 |
| Fitting | C32×#338 | C3 |
| socle | C2×C32 | #95⋊C3 |
| prime radical | C23×C32 | SL(2,𝔽3) |
| cosocle | C3×#338 | C32 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C32×C4 | |
142 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1292);