Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | PSU(3,𝔽2)⋊C16 |
| Sylow | C32 ⋊ Q8⋊C16 |
108 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^7,3^2|a^2=d,b^2=e,c^2=e,d^2=f,f^2=g,b^a=bc,c^a=ce,h^a=hi,i^a=i^2,c^b=ce,h^b=i,i^b=h^2,h^c=h^2i^2,i^c=h^2i,h^e=h^2,i^e=i^272 conjugacy classes — view the full character table.
| maximal subgroup of | #72218959, C3×PSU(3,𝔽2)⋊C16, C16×AΓL(1,𝔽9), … |
| maximal normal subgroup of | C3×PSU(3,𝔽2)⋊C16, C16×AΓL(1,𝔽9), #72245163 |
| maximal quotient of | C3×PSU(3,𝔽2)⋊C16, #72219099 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 5702);