Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #88401⋊C3 |
| Fitting | D4×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 | D4×C27⋊C3 | C2 |
| socle | C2×C3 | C22×C3×D9 |
| hypercenter | D4 | D27⋊C3 |
| hypocenter | C27 | C3×#41 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ D4 | |
176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3,2,3^3|b^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, 1066);