Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #1094⋊Q8 |
| unambiguous semidirect product | ((C22⋊C9)⋊Q8)⋊C3 |
157 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3,2,3^2,2^2|a^2=d,b^2=d,e^3=f,b^a=bd,e^a=e^2f^2,f^a=f^2,g^a=h,h^a=g,e^c=ef^2,g^e=h,h^e=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #10589500⋊C3, #10589503⋊C3, #10589516⋊C3, … |
| maximal normal subgroup of | #10589500⋊C3, #10589503⋊C3, #10589516⋊C3, … |
| maximal quotient of | #10589488⋊C3, #10589491⋊C3, #10589493⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2300);