Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Sylow | C34 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10620145 |
| commutator, Fitting, cosocle | C34×C4 | C22 |
| Frattini, hypercenter | C4 | D3×C3⋊(C3⋊D3) |
| socle | C2×C34 | D4 |
| prime radical | C33×D12 | C2 |
| hypocenter | C34 | SD16 |
| LCS | G ▷ C34×C4 ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C4 | |
872 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^4,3^4|b^2=d,c^2=d,b^a=bc,c^a=cd,h^a=h^2,c^b=cd,e^b=e^2,f^b=f^2,g^b=g^2129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2768);