Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D9⋊(C22⋊A4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | Dic9⋊(C22⋊A4) |
| derived | C24×C9 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9⋊(C22⋊A4) |
| commutator, hypocenter | C24×C9 | C3×C4 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| Fitting, cosocle | C25×C9 | C2×C3 |
| socle | C25×C3 | C3×D3 |
| prime radical | C2×C3×C22⋊A4 | D3 |
| LCS | G ▷ C24×C9 | |
| UCS | C1 ◁ C2 | |
800 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 54 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:
2,3,2^5,3^2|a^2=c,h^3=i,h^a=h^2i^2,i^a=i^2,d^b=de,e^b=d,f^b=fg,g^b=f,h^b=hi^280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30717);