Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of (C22⋊C81)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C81)⋊C4 |
| derived | C22⋊C81 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C22⋊C81)⋊C2 |
| commutator, hypocenter | C22⋊C81 | C4 |
| Frattini | C2×C27 | S4 |
| Fitting | C23×C27 | D3 |
| socle, prime radical | C23×C3 | D27 |
| cosocle | C2×C22⋊C81 | C2 |
| LCS | G ▷ C22⋊C81 | |
| UCS | C1 ◁ C2 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^2,3^4,2^2|a^2=b,c^3=d,d^3=e,e^3=f,c^a=c^2d^2e^2f^2,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2,g^a=gh,g^c=h,h^c=gh114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 59);