Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ C3×#5429 |
334 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,18,12,2,20,21,6,14,22,5,10,13)(3,16,19,9,4,17,11,8,24,7,15,23)(26,29,30,27,28,31);(1,2,3,7,6,5,11,9)(4,12,10,19,15,22,20,24)(8,17,14,21,16,23,18,13)(25,26,28,31,30,29,27)Polycyclic presentation:
2^2,3,2,3,7,2^3|a^2=d,b^2=i,d^2=i,g^2=i,h^2=i,b^a=bdi,d^b=di,e^a=e^2,f^b=f^6,f^c=f^2,g^a=gi,g^e=h,h^a=gh,h^e=ghi,h^g=hi67 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5429 |
| maximal normal subgroup of | AGL(1,𝔽7)×#5429 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 3, 7, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);