Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D9⋊(#120⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | D9⋊(#120⋊C3) |
| unambiguous semidirect product | (D9⋊(C22⋊A4))⋊C2 |
| Fitting | C9×#120 ⋊ C2×C3 |
| derived | C24×C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C9×#120 | C2×C3 |
| LCS | G ▷ C24×C9 | |
| UCS | C1 | |
434 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 434 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) = 72 |
Polycyclic presentation:
2^2,3,2^4,3^2|h^3=i,h^a=h^2i^2,i^a=i^2,d^b=df,e^b=efg,d^c=de,e^c=d,f^c=g,g^c=fg,h^c=hi^244 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13176);