Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D27⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic27⋊A4 |
| unambiguous semidirect product | (C27⋊A4)⋊C4 |
| derived | C22×C27 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D27⋊A4 |
| commutator, hypocenter | C22×C27 | C3×C4 |
| Frattini | C2×C9 | D3×A4 |
| Fitting, cosocle | C23×C27 | C2×C3 |
| socle | C23×C3 | C3×D9 |
| prime radical | C2×C3×A4 | D9 |
| LCS | G ▷ C22×C27 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2,3,2^3,3^3|a^2=c,f^3=g,g^3=h,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,d^b=de,e^b=d,f^b=fh72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 750);