Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#33348 ⋊ C2 |
| Sylow | C3 ⋊ #493474 |
5616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(9,18)(17,19);(1,2,3)(6,12)(9,17)(13,16)(18,19);(4,6,7,13)(5,9,10,18)(8,16)(11,19)(12,14)(15,17);(4,5)(6,9)(7,10)(8,15)(11,14)(12,19)(13,18)(16,17)Polycyclic presentation:
2^9,3|b^2=e,f^2=i,c^a=ch,c^b=cefi,d^a=dh,d^b=dg,e^c=eh,f^a=fi,f^b=fh,f^c=fh,f^e=fi,g^a=gi,h^b=hi,j^d=j^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #72271853, #72271892, D3×#493474 |
| maximal normal subgroup of | #72271853, #72271892, D3×#493474 |
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, 2, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);