Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #36066 ⋊ D3 |
1266 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,5,7,6)(9,26,25,10,27,30,12,23,32,15,19,21)(11,22,24,18,13,29,17,16,31,14,20,28);(3,8)(5,6);(1,2)(3,5)(4,7)(6,8)(10,14)(11,17)(13,21)(15,18)(16,24)(19,28)(20,30)(22,31)(23,25)(26,32)(27,29)Polycyclic presentation:
2^6,3,2^3|c^2=ej,d^2=f,h^2=j,i^2=j,c^a=cd,c^b=ce,d^a=df,d^b=dfj,d^c=dj,e^a=efj,g^a=g^2,h^a=ij,h^g=hi,i^a=hj,i^g=h,i^h=ij61 conjugacy classes — view the full character table.
| maximal quotient of | #72258536 |
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, 2, 2, 2, 3, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 7, F.8);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);