Finite Group DatabasebetaFinite Group Databasebeta| order | 15552 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
2076 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 2076 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4)(6,7,8)(9,12,10,11)(13,19,20,15,14,21,18,16,17)(22,23)(24,30)(26,27)(28,29);(5,8)(6,7)(10,12,11)(14,16,19)(17,21,20)(22,28,29,25,26,30,23,24,27)Polycyclic presentation:
2,3^2,2,3,2^2,3^2,2^2|a^2=d,c^3=e,h^3=i,c^a=ci^2,c^b=ce,f^c=fg,g^c=f,h^a=h^2i^2,h^b=hi,i^a=i^2,j^a=k,j^h=k,k^a=j,k^h=jk264 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", "k");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.11);;
gap> SetConjugate(coll, 10, 8, F.11);;
gap> SetConjugate(coll, 11, 1, F.10);;
gap> SetConjugate(coll, 11, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);