Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | C4⋊(C3⋊((C22⋊C9)⋊C2)) |
304 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^3,2^2|b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^2,f^a=f^2,g^a=h,h^a=g,g^d=h,h^d=gh78 conjugacy classes — view the full character table.
| maximal subgroup of | #13146269, #13146270, #13146276, … |
| maximal normal subgroup of | #13146269, #13146270, #13146276, … |
| maximal quotient of | C4⋊(C3⋊#3688), C4⋊(C3⋊#3689), #13146999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2346);