Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
4547 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 4547 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4)(10,12)(13,21,16,15,14,18,19,17,20);(2,3)(6,8)(9,10)(11,12)(14,16)(15,19)(17,20)(18,21);(5,6)(7,8)Polycyclic presentation:
2^3,3,2^2,3^2,2^2|g^3=h,b^a=be,c^b=cf,g^b=g^2h^2,g^d=gh,h^b=h^2,i^b=j,i^g=j,j^b=i,j^g=ij196 conjugacy classes — view the full character table.
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, 3, 2, 2, 3, 3, 2, 2]);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 7, F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);