Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C33 |
No aliases for this group.
| Fitting | C33×#120 ⋊ C2 |
| Sylow | C33 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×#120 | C2 |
| hypercenter | #120 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #538 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #120 | |
3500 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^6,3^3|c^2=e,b^a=bf,d^a=def,g^a=g^2,h^a=h^2,i^a=i^2,c^b=cf,d^c=de204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45781);