Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #376⋊C27 |
| Sylow | #376 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | C42⋊C3 |
| commutator, hypocenter | #95 | C2×C27 |
| Frattini | C23×C9 | C2×A4 |
| Fitting | C9×#376 | C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C9 |
| cosocle | C9×#95 | C2×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C4×C9 | |
113 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|a^2=gh,b^3=c,c^3=d,e^2=g,f^2=gi,e^b=ef,f^b=ehi,g^b=ghi,h^b=i,i^b=hi,f^e=fgh216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 199);