Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C34 |
| unambiguous semidirect product | C4⋊#66472 |
| Fitting | C34×C4 ⋊ C4 |
| Sylow | C34 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#66472 |
| commutator, socle | C2×C34 | C2×C4 |
| Fitting | C34×C4 | C4 |
| prime radical, cosocle | C2×C3⋊(C3⋊(C3⋊D3)) | C22 |
| hypercenter | C4 | #66472 |
| hypocenter | C34 | C4⋊C4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C4 | |
956 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^4|a^2=c,b^2=d,b^a=bd,e^a=ef,f^a=ef^2,g^a=gh,h^a=gh^2,e^c=e^2,f^c=f^2,g^c=g^2,h^c=h^290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3318);