Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊S4)) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C36 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊SL(2,ℤ/4))) |
| unambiguous semidirect product | A4⋊(C3⋊(C3⋊Dic3)) |
| derived | C33×A4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C3⋊(C3⋊S4)) |
| commutator, hypocenter | C33×A4 | C4 |
| Fitting, socle | C23×C33 | D3 |
| prime radical, cosocle | C2×C33×A4 | C2 |
| LCS | G ▷ C33×A4 | |
| UCS | C1 ◁ C2 | |
1468 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^2,3^4,2^2|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^2,g^a=gh,g^c=h,h^c=gh114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3284);