Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
1402 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 108 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,4,3,2)(5,6,8,7)(10,12)(14,16)(15,19)(17,20)(18,21);(6,7,8)(9,10)(11,12)(13,20,21,15,14,16,19,18,17);(2,3,4)(6,7,8)Polycyclic presentation:
2^3,3^3,2^4|e^3=f,b^a=bc,d^b=d^2,e^b=e^2f^2,f^b=f^2,g^b=h,g^d=h,g^e=h,h^b=g,h^d=gh,h^e=gh,i^b=gij,i^d=j,i^e=ghi,j^b=ghj,j^d=ij,j^e=gj108 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, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.7*F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);