Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #395⋊C27 |
| Sylow | #395 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C2×C27 |
| Frattini | C23×C9 | C2×A4 |
| Fitting | C9×#395 | C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C9 |
| cosocle | C22×Q8×C9 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C9 | |
121 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,b^3=c,c^3=d,e^2=i,f^2=i,e^a=eg,f^a=fgh,e^b=ef,f^b=e,g^b=h,h^b=gh,f^e=fi162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 205);