Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C34 ⋊ C2×#106 |
616 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,3^4|a^2=e,b^2=d,d^2=f,b^a=bd,d^a=df,g^a=g^2,g^b=h,g^c=g^2,g^d=gh^2,g^f=g^2,h^a=gh,h^b=gh^2,h^c=h^2,h^d=g^2h^2,h^f=h^2,i^a=gh^2i^2,i^b=j,i^c=ghi,i^d=ij^2,i^f=i^2,j^a=h^2ij,j^b=ij^2,j^c=gj,j^d=i^2j^2,j^f=j^254 conjugacy classes — view the full character table.
| maximal subgroup of | #72220811, #72222921, #72252826 |
| maximal normal subgroup of | #72220811, #72222921, #72252826 |
| maximal quotient of | #72252820 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 6, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.7*F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.8^2*F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 3, F.7*F.10);;
gap> SetConjugate(coll, 10, 4, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 6, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);