Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#8534 ⋊ C2 |
| Sylow | C3 ⋊ #144386 |
3576 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 3576 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(4,6,7,11)(5,9)(8,10)(12,14,19,17)(13,16,18,15);(1,2,3)(4,6,7,11)(5,8,10,9)(12,13)(14,15)(16,17)(18,19);(4,5,7,10)(6,9)(8,11)(14,18)Polycyclic presentation:
2^9,3|a^2=d,b^2=g,c^2=defgi,f^2=i,b^a=bde,c^a=ceg,c^b=cdefi,d^b=dg,d^c=dg,e^a=eg,e^b=eg,e^c=eh,f^a=fgh,f^c=fgh,f^e=fi,h^b=hi,h^c=hi,j^c=j^284 conjugacy classes — view the full character table.
| maximal subgroup of | #72253601, D3×#144386, #72270536 |
| maximal normal subgroup of | #72253601, D3×#144386, #72270536 |
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.7);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.7*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);