Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#401⋊C9) |
| unambiguous semidirect product | (C3⋊#401)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(#126⋊C3) |
| commutator, hypocenter | C3×#95 | C2×C9 |
| Frattini | C23×C3 | D3×A4 |
| Fitting, cosocle | C32×#95 | C2×C3 |
| socle, prime radical | C23×C32 | C2×A4 |
| LCS | G ▷ C3×#95 | |
| UCS | C1 ◁ C2×C3 | |
137 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^2,2^2,3,2^3|a^2=gh,b^3=c,d^2=g,e^2=gi,d^a=dhi,e^a=ei,f^a=f^2,d^b=de,e^b=dhi,g^b=ghi,h^b=i,i^b=hi,e^d=egh90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2887);