Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C34 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620206 |
| commutator, cosocle | C3⋊(C3⋊(C3⋊Dic3)) | C22 |
| Fitting, socle, prime radical | C2×C34 | D4 |
| hypocenter | C34 | Q16 |
| LCS | G ▷ C3⋊(C3⋊(C3⋊Dic3)) ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
351 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3^4|a^2=d,b^2=d,c^2=d,b^a=bcd,c^a=cd,e^a=e^2,g^a=g^2,h^a=g^2h,c^b=cd,e^b=f^2,f^b=e^2,g^b=g^2h^2,e^c=e^2,f^c=f^2,g^c=g^2,h^c=h^251 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3177);