Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C81 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C22⋊(C22⋊C81) |
| derived, Sylow | C24 ⋊ C81 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C27 | C22⋊A4 |
| commutator, hypocenter | C24 | C81 |
| Fitting, cosocle | C24×C27 | C3 |
| socle, prime radical | C24×C3 | C27 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C27 | |
115 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
3^4,2^4|a^3=b,b^3=c,c^3=d,e^a=ef,f^a=e,g^a=gh,h^a=g216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 632);