Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #396⋊C27 |
| derived | #95 ⋊ C2×C27 |
| Sylow | #396 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #126⋊C3 |
| commutator, hypocenter | #95 | C2×C27 |
| Frattini | C23×C9 | C2×A4 |
| Fitting | C9×#396 | C3 |
| socle | C23×C3 | C2×C22⋊C9 |
| prime radical | C24×C3 | C22⋊C9 |
| cosocle | C9×#95 | C2×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2×C9 | |
129 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3^3,2^5|b^3=c,c^3=d,e^2=g,f^2=gi,e^a=ehi,f^a=fi,e^b=ef,f^b=ehi,g^b=ghi,h^b=i,i^b=hi,f^e=fgh162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 206);