Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C81 ⋊ C2 |
| Sylow | C81 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D81 |
| commutator, cosocle | C2×C81 | C23 |
| Frattini | C2×C27 | C22×D3 |
| Fitting | Q8×C81 | C2 |
| socle | C2×C3 | C22×D27 |
| hypercenter | Q8 | D81 |
| hypocenter | C81 | #41 |
| LCS | G ▷ C2×C81 ▷ C81 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2^4,3^4|b^2=d,c^2=d,e^3=f,f^3=g,g^3=h,c^a=cd,e^a=e^2f^2g^2h^2,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=cd210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 70);