Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#41⋊C27) |
| unambiguous semidirect product | #239⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C9 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C2×C27 |
| Fitting, cosocle | C3×Q8×C9 | C2×C3 |
| socle | C2×C32 | C2×C22⋊C9 |
| prime radical | C3×#239 | C9 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C9 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2,3^3,2^2,3,2|a^2=h,b^3=c,c^3=d,e^2=h,f^2=h,g^a=g^2,e^b=ef,f^b=e,f^e=fh189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 751);