Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C22⋊(C42⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | C22⋊(C42⋊C27) |
| derived, Sylow | C22×C42 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C22⋊(C42⋊C3) |
| commutator, hypocenter | C22×C42 | C27 |
| Frattini | C22×C9 | C22⋊A4 |
| Fitting, cosocle | C22×C42×C9 | C3 |
| socle, prime radical | C24×C3 | C22⋊C9 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C9 | |
285 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
3^3,2^6|a^3=b,b^3=c,f^2=h,g^2=hi,d^a=de,e^a=d,f^a=fg,g^a=fhi,h^a=i,i^a=hi216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2262);