Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C8×C81 ⋊ C2 |
| Sylow | C81 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D324 |
| commutator, cosocle | C4×C81 | C22 |
| Frattini | C4×C27 | C2×D3 |
| Fitting | C8×C81 | C2 |
| socle | C2×C3 | D108 |
| prime radical | D324 | C2 |
| hypercenter | C8 | D81 |
| hypocenter | C81 | SD16 |
| LCS | G ▷ C4×C81 ▷ C2×C81 ▷ C81 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2^4,3^4|b^2=c,c^2=d,e^3=f,f^3=g,g^3=h,b^a=bc,c^a=cd,e^a=e^2f^2g^2h^2,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2327 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 7);