Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (C22⋊A4)⋊C27 |
| derived, Sylow | C24 ⋊ C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C9 | A42 |
| commutator, hypocenter | C24 | C3×C27 |
| Fitting, cosocle | C24×C9 | C32 |
| socle | C24×C3 | C3×C9 |
| prime radical | C3×C22⋊A4 | C9 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C9 | |
101 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
3^4,2^4|a^3=c,b^3=c,c^3=d,g^a=gh,h^a=g,e^b=ef,f^b=e144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1792);