Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊#4628) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | C3⋊(C3⋊S4) |
| commutator, hypocenter | C32×SL(2,𝔽3) | C8 |
| Fitting | C32×#131 | D3 |
| socle | C2×C32 | A4⋊C8 |
| prime radical | C32×#41⋊C3 | C4 |
| cosocle | C32×#131⋊C3 | C2 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C8 | |
318 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^3,3^3,2^3|a^2=b,b^2=c,c^2=i,g^2=i,h^2=i,d^a=d^2,e^a=e^2,f^a=f^2,g^a=gh,h^a=hi,g^d=h,h^d=ghi,h^g=hi144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23085);