Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C27⋊C3 ⋊ C2 |
| Sylow | C27⋊C3 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D27⋊C3 |
| commutator, cosocle | C2×C27 | C23×C3 |
| Frattini | C2×C9 | C22×C3×D3 |
| Fitting | Q8×C27⋊C3 | C2 |
| socle | C2×C3 | C22×C3×D9 |
| hypercenter | Q8 | D27⋊C3 |
| hypocenter | C27 | C3×#41 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3,2,3^3|b^2=e,c^2=e,f^3=g,g^3=h,c^a=ce,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=ce,f^d=fh105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1092);