Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22×C32 |
No aliases for this group.
| Sylow | C24 ⋊ C32⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | A42 |
| commutator | C24×C3 | C3×C9 |
| Fitting, socle, cosocle | C24×C32 | C32 |
| prime radical | C32×C22⋊A4 | C3 |
| hypocenter | C24 | C32⋊C9 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C32 | |
200 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
3^4,2^4|a^3=d,b^3=d,b^a=bc,g^a=gh,h^a=g,e^b=ef,f^b=e96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1797);