Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D9×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C9⋊S4)⋊C8 |
| derived | C9×A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D9×S4 |
| commutator, hypocenter | C9×A4 | C2×C8 |
| Frattini | C3×C4 | D3×S4 |
| Fitting | C22×C4×C9 | C2×D3 |
| socle | C23×C3 | #624 |
| prime radical | C2×C9⋊S4 | C4 |
| cosocle | C4×C9×A4 | C22 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 ◁ C4 | |
269 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=c,b^2=c,c^2=d,f^3=g,f^a=f^2g^2,g^a=g^2,e^b=e^2,h^b=hi,h^e=i,i^e=hi120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12252);