Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D27⋊SL(2,𝔽3) |
| Fitting, derived | Q8×C27 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D27⋊A4 |
| commutator, Fitting, cosocle, hypocenter | Q8×C27 | C2×C3 |
| Frattini | C2×C9 | D3×A4 |
| socle | C2×C3 | A4×D9 |
| LCS | G ▷ Q8×C27 | |
| UCS | C1 ◁ C2 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3,2^2,3,2,3^2|c^2=f,d^2=f,e^3=g^2,g^3=h^2,e^a=e^2gh,g^a=g^2h,h^a=h^2,c^b=d,d^b=cd,e^b=eh,d^c=df57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 748);