Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #13049186 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | C3×H(3,𝔽3) ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | C3×#1088 | C22 |
| Frattini | C3 | #89092 |
| Fitting, hypocenter | C3×H(3,𝔽3) | SD16 |
| socle | C32 | AΓL(1,𝔽9) |
| prime radical | #1088⋊D3 | C2 |
| LCS | G ▷ C3×#1088 ▷ C3×#272 ▷ C3×H(3,𝔽3) | |
| UCS | C1 | |
116 conjugacy classes of subgroups.
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
2^4,3^4|b^2=d,c^2=d,b^a=bcd,c^a=cd,e^a=e^2fh,f^a=fh,g^a=g^2,h^a=h^2,c^b=cd,e^b=f^2,f^b=eh^2,e^c=ef^2,f^c=e^2f^2h^2,e^d=e^2h,f^d=f^2h^2,f^e=fh^230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2904);