Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
No aliases for this group.
578 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 35 points:
(2,4)(5,7,8)(9,35,13,19,15,16,18,17,22,12,25,24,33,27,28,20,29,34,21,31,23,11,10,30,32,14,26);(1,2)(3,4)(6,7)(10,13)(11,18)(12,21)(14,22)(15,23)(16,25)(17,26)(19,32)(20,33)(24,27)(28,35)(29,34)(30,31)Polycyclic presentation:
2^2,3,2,3^4,2^2|c^3=e,f^3=g,g^3=h,b^a=bd,f^a=f^2g^2h^2,f^c=fgh^2,f^e=fh,g^a=g^2h^2,g^c=gh,h^a=h^2,i^a=j,i^f=ij,j^a=i,j^f=i190 conjugacy classes — view the full character table.
| maximal subgroup 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, 2, 3, 2, 3, 3, 3, 3, 2, 2]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);