Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×#1891 ⋊ C2 |
| Sylow | C3 ⋊ #147782 |
2412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,4)(5,8)(6,14)(7,17)(9,12)(10,20)(11,13)(15,18)(16,19)(21,22,23);(2,4)(5,7,8,19,12,17,9,16)(6,13,15,11,14,10,18,20)(22,23);(1,2,3,4)(5,6,10,16,12,14,13,19)(7,8,18,20,17,9,15,11)Polycyclic presentation:
2^9,3|a^2=e,b^2=gh,e^2=i,f^2=i,g^2=i,b^a=begi,c^a=cefi,c^b=cd,d^b=dh,f^b=fh,g^b=gi,g^c=gh,g^d=gi,g^f=gi,h^a=hi,h^b=hi,j^b=j^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #72248020, #72253930, D3×#147782, … |
| maximal normal subgroup of | #72248020, #72253930, D3×#147782, … |
| maximal quotient of | #72271617 |
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.5);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);