Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11724023 |
| commutator, cosocle, hypocenter | #63703 | C2×C3 |
| Frattini, socle | C2×C3 | #1020⋊C3 |
| Fitting | C3×H(5,𝔽2) | C3×D3 |
| prime radical | C3×#997 | D3 |
| LCS | G ▷ #63703 | |
| UCS | C1 ◁ C2 | |
99 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,c^3=d,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd^2,c^a=c^2d^2,d^a=d^2,e^a=gi,f^a=hi,g^a=ei,h^a=fi,e^b=efi,f^b=e,g^b=gh,h^b=g,e^c=f,f^c=efi,g^c=gh,h^c=g,f^e=fi,h^g=hi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30116);