Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | Q8⋊(Q8⋊C27) |
| derived, Sylow | Q82 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22×C9 | C22⋊A4 |
| commutator, hypocenter | Q82 | C27 |
| Fitting, cosocle | Q82×C9 | C3 |
| socle, prime radical | C22×C3 | C22⋊(C22⋊C9) |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C22×C9 | |
154 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1728 |
Polycyclic presentation:
3^3,2^6|a^3=b,b^3=c,d^2=i,e^2=i,f^2=h,g^2=h,d^a=de,e^a=d,f^a=fg,g^a=f,e^d=ei,g^f=gh171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2264);