Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C33×C4 |
No aliases for this group.
| Fitting | C33×#117 ⋊ C2 |
| Sylow | C33 ⋊ #555 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×#117 | C2 |
| hypercenter | #117 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #555 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #117 | |
3304 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=f,c^2=e,d^2=f,c^a=cf,g^a=g^2,h^a=h^2,i^a=i^2,d^b=de285 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45707);