Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#15258 ⋊ C2 |
| Sylow | C3 ⋊ #144386 |
3576 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 3576 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,3)(2,5,6,8)(4,7)(9,10,11,16)(12,13,15,14)(18,19);(5,8)(10,14,13,16)(11,15)(17,18,19);(1,2)(3,5,7,8)(4,6)(10,13)Polycyclic presentation:
2^9,3|a^2=d,b^2=gh,c^2=degh,b^a=bdgh,c^a=cf,c^b=cde,d^c=dg,e^a=eh,e^b=ei,e^c=egi,f^a=fg,f^b=fgh,f^c=fh,f^e=fi,h^c=hi,j^c=j^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #72255074, #72255279, D3×#144386 |
| maximal normal subgroup of | #72255074, #72255279, D3×#144386 |
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, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.4*F.5*F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);