Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(#3688⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(((C22⋊C9)⋊C2)⋊C3) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C2×C3 |
| Frattini | C2×C3 | C3×C3⋊S4 |
| Fitting | C33×Q8 | D3 |
| socle | C2×C32 | C3×S4 |
| prime radical | C2×C33 | S4 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
171 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2,3^4,2^3|a^2=h,c^3=e,f^2=h,g^2=h,c^a=c^2e^2,d^a=d^2,e^a=e^2,f^a=fg,g^a=gh,c^b=ce,f^c=g,g^c=fgh,g^f=gh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2121);