Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C9⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C9 |
| unambiguous semidirect product | C9⋊((C22⋊C9)⋊C4) |
| unambiguous semidirect product | (C22⋊C9)⋊Dic9 |
| derived | C9×C22⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊((C22⋊C9)⋊C2) |
| commutator, hypocenter | C9×C22⋊C9 | C4 |
| Frattini | C2×C32 | C3⋊S4 |
| Fitting | C23×C3×C9 | D3 |
| socle, prime radical | C23×C32 | C3⋊D3 |
| cosocle | C2×C9×C22⋊C9 | C2 |
| LCS | G ▷ C9×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
208 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^2,3^4,2^2|a^2=b,c^3=f,d^3=e,c^a=c^2f^2,d^a=d^2e^2,e^a=e^2,f^a=f^2,g^a=gh,g^c=h,h^c=gh114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 731);