Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C3⋊#10593123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D3×C22⋊A4 |
| commutator, hypocenter | C22×C3×Q8 | C2×C9 |
| Fitting, cosocle | C22×C32×Q8 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| prime radical | C22×C3×#239 | C3 |
| LCS | G ▷ C22×C3×Q8 | |
| UCS | C1 ◁ C2×C3 | |
709 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 709 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^4,3,2|a^2=i,b^3=c,f^2=i,g^2=i,h^a=h^2,d^b=de,e^b=d,f^b=fg,g^b=f,g^f=gi108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30718);