Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #543⋊C27 |
| derived, Sylow | #543 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22×C9 | C22⋊A4 |
| commutator, hypocenter | #543 | C27 |
| Fitting, cosocle | C9×#543 | C3 |
| socle | C22×C3 | C22⋊(C22⋊C9) |
| prime radical | C24×C3 | C22⋊C9 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C9 | |
178 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
3^3,2^6|a^3=b,b^3=c,f^2=h,g^2=h,d^a=de,e^a=d,f^a=fg,g^a=f,g^d=gi,f^e=fi,g^f=gh162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2263);