Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Sylow | C3 ⋊ #126544 |
708 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)(12,13,14,19,15,20,22,27,16,18,21,26,23,17,24,25);(4,5,6,9,7,8,10,11);(2,3)(5,8)(9,11)(13,17,18,20)(14,21)(19,27,26,25)(22,24)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=bg,c^a=chi,d^c=di,f^a=fi,j^a=j^2288 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#126544, #72225880, #72225906, … |
| maximal normal subgroup of | D3×#126544, #72225880, #72225906, … |
| maximal quotient of | #72226216 |
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.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);