Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D3×D9 |
| commutator, cosocle | C23×C3×C9 | C23 |
| Frattini | C23×C3 | C2×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C24 | D3×D9 |
| hypocenter | C3×C9 | #392 |
| LCS | G ▷ C23×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C23 ◁ C24 | |
1202 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|b^2=f,g^3=i,b^a=bf,c^a=ce,g^a=g^2i^2,i^a=i^2,c^b=cd,h^b=h^2186 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7203);