Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C9⋊#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C9⋊#3708 |
| derived, Sylow | C32×C9 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C9⋊#157 |
| commutator, hypocenter | C32×C9 | C16 |
| Frattini | C3×C4 | C3⋊#157 |
| Fitting | C32×C4×C9 | C4 |
| socle, prime radical | C2×C33 | C3⋊C8 |
| cosocle | C9×C3⋊(C3⋊C8) | C2 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C4 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3^4|a^2=b,b^2=c,c^2=d,g^3=h,e^a=ef,f^a=ef^2,g^a=g^2h^2,h^a=h^2,e^b=e^2,f^b=f^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1240);