Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C27⋊(C22⋊A4) |
| Fitting | C24×C27 ⋊ C3 |
| Sylow | C24 ⋊ C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C3×C22⋊A4 |
| commutator, socle | C24×C3 | C3×C9 |
| Fitting | C24×C27 | C3 |
| prime radical | C3×C22⋊A4 | C9 |
| cosocle | C24×C9 | C32 |
| hypercenter | C27 | C22⋊A4 |
| hypocenter | C24 | C27⋊C3 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C9 ◁ C27 | |
164 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
3^4,2^4|b^3=c,c^3=d,b^a=bd,e^a=ef,f^a=e,g^a=gh,h^a=g168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1883);