Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊C64)) |
| derived, Sylow | C33 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | C3⋊(C3⋊D3) |
| commutator, hypocenter | C33 | C64 |
| Fitting, cosocle | C33×C32 | C2 |
| socle, prime radical | C2×C33 | C32 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C32 | |
196 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1728 |
Polycyclic presentation:
2^6,3^3|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^2,h^a=h^2,i^a=i^2480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2712);