Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Sylow | C3 ⋊ #135758 |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,13,14,17,15,19,20,24,16,18,21,23,22,25,27,26);(4,5,6,8,7,9,10,11);(2,3)(5,8)(6,10)(9,11)(13,17,18,23)(14,20)(15,22)(19,26,25,24)(21,27)Polycyclic presentation:
2^9,3|a^2=d,b^2=e,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,d^c=di,e^a=eg,f^a=fh,h^a=hi,j^a=j^2252 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#135758, #72225893, #72225906, … |
| maximal normal subgroup of | D3×#135758, #72225893, #72225906, … |
| maximal quotient of | #72226200 |
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.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);