Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#8101 ⋊ C2 |
| Sylow | C3 ⋊ #115204 |
5622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(4,6,7,11)(5,9,10,8)(12,13,15,17)(14,16,19,18);(1,2,3)(4,6,7,11)(5,8)(9,10)(12,14)(13,16)(15,19)(17,18);(4,5,7,10)(6,8)(9,11)(12,13,15,17)(14,18)(16,19)Polycyclic presentation:
2^9,3|a^2=f,b^2=dfg,c^2=h,b^a=bd,c^a=ce,c^b=cdfghi,d^a=dh,d^b=dhi,d^c=dh,e^a=ehi,e^b=ei,f^b=fh,f^c=fhi,g^a=gi,g^b=gi,g^c=gi,j^c=j^2114 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#115204, #72248860, #72253433 |
| maximal normal subgroup of | D3×#115204, #72248860, #72253433 |
| maximal quotient of | #72253679, #72271768 |
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.6);;
gap> SetPower(coll, 2, F.4*F.6*F.7);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);