Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C9×C16 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊(C3⋊C16))⋊C9 |
| derived | C32 ⋊ C9×C16 |
| Sylow | C32⋊C9 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C9×C16 |
| Frattini | C32×C8 | C3×D3 |
| Fitting | C8×C32⋊C9 | C2 |
| socle | C2×C32 | C3×C3⋊C8 |
| prime radical | C2×C33 | C3×C8 |
| cosocle | C33×C8 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C8 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3,2,3,2^2,3^2|a^2=c,b^3=d,c^2=e,e^2=f,g^a=g^2,h^a=h^2,g^b=gh240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 29);