Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C32⋊C9 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | C3⋊D9 |
| commutator, hypocenter | C32⋊C9 | C16 |
| Frattini | C32×C8 | C3⋊D3 |
| Fitting, cosocle | C8×C32⋊C9 | C2 |
| socle | C2×C32 | C3⋊(C3⋊C8) |
| prime radical | C2×C33 | C3⋊C8 |
| LCS | G ▷ C32⋊C9 | |
| UCS | C1 ◁ C3×C8 | |
103 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3^4|a^2=b,b^2=c,c^2=d,f^3=h,e^a=e^2,f^a=f^2h^2,h^a=h^2,f^e=fg168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 42);