Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C3 |
No aliases for this group.
| Fitting | C2×C33×Q8 ⋊ C22 |
| Sylow | C33 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C2×C33×Q8 | C22 |
| hypercenter | C2×Q8 | D3×C3⋊D3 |
| hypocenter | C33 | #585 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
5444 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|c^2=f,d^2=f,e^a=ef,g^a=g^2,h^a=h^2,d^b=df,i^b=i^2,d^c=df171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47697);