Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C81 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #41⋊C81 |
| Sylow | #41 ⋊ C81 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C27 | A4 |
| commutator, hypocenter | Q8 | C2×C81 |
| Frattini | C2×C27 | C2×A4 |
| Fitting | #41×C27 | C3 |
| socle | C2×C3 | C2×C22⋊C27 |
| prime radical | C3×#41 | C27 |
| cosocle | Q8×C27 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C4×C27 | |
45 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2,3^4,2^3|a^2=h,b^3=c,c^3=d,d^3=e,f^2=h,g^2=h,f^b=fg,g^b=f,g^f=gh378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 62);