Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C9⋊#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C9⋊#5936 |
| derived | C9×C22⋊A4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊#1020 |
| commutator, hypocenter | C9×C22⋊A4 | C4 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting | C25×C9 | D3 |
| socle | C25×C3 | C3⋊D3 |
| prime radical | C2×C3×C22⋊A4 | D3 |
| cosocle | C2×C9×C22⋊A4 | C2 |
| LCS | G ▷ C9×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
518 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 518 conjugacy classes of subgroups — too many to draw. Only its 45 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:
2^2,3^3,2^4|a^2=b,d^3=e,c^a=c^2,d^a=d^2e^2,e^a=e^2,f^a=fg,h^a=hi,f^c=g,g^c=fg,h^c=i,i^c=hi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30609);