Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C33 |
No aliases for this group.
| Fitting | C2×C33×#41 ⋊ C2 |
| Sylow | C33 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C25 |
| Frattini | C2 | C24×C3⋊(C3⋊D3) |
| Fitting | C2×C33×#41 | C2 |
| hypercenter | C2×#41 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #585 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
11340 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^6,3^3|c^2=f,e^a=ef,g^a=g^2,h^a=h^2,i^a=i^2,d^b=df294 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47826);