Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D9⋊(#131⋊C3) |
| Fitting | C9×#131 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D9⋊A4 |
| commutator, hypocenter | Q8×C9 | C2×C3×C4 |
| Frattini | C3×C4 | C2×D3×A4 |
| Fitting | C9×#131 | C2×C3 |
| socle | C2×C3 | C4×D3×A4 |
| prime radical | D9⋊(#41⋊C3) | C2 |
| cosocle | C9×#41 | C22×C3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C8 | |
235 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3,2^3,3,2,3|b^2=d,d^2=h,e^2=h,f^2=h,g^3=i,g^a=g^2i^2,i^a=i^2,e^c=ef,f^c=e,g^c=gi^2,f^e=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13202);