Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| derived | (C22⋊A4)⋊C9 ⋊ C2×C9 |
658 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,4,3,2)(5,8,6,7)(9,13,17,11,16,12,14,10,15)(19,25,26,21,22,23)(20,24);(6,7,8)(18,22,21,20,26,23,24,19,25);(2,3,4)(6,7,8)Polycyclic presentation:
2,3^5,2^4|b^3=c,e^3=f,d^a=d^2,e^a=e^2f^2,e^b=ef,f^a=f^2,g^d=h,g^e=h,h^a=gh,h^d=gh,h^e=gh,i^a=hij,i^d=j,i^e=ghi,j^a=gj,j^d=ij,j^e=gj150 conjugacy classes — view the full character table.
| maximal quotient of | #72259339 |
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, 3, 3, 3, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.8*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, 1, F.7*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);