Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #394 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C23×C33 | C23 |
| Fitting | C24×C33 | C22 |
| prime radical | C22×C32×#67 | C2 |
| hypercenter | C24 | D3×C3⋊D3 |
| hypocenter | C33 | #394 |
| LCS | G ▷ C23×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C24 | |
1956 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^a=be,c^a=cd,g^a=g^2,h^a=h^2,c^b=cf,i^b=i^2186 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 27494);