Subsections

• Accessing Group Information

The functions in this group provide access to basic information stored for a permutation group G.

G . i : GrpPerm, RngIntElt -> GrpPermElt

The i-th defining generator for G. A negative subscript indicates that the inverse of the generator is to be created. G.0 is Identity(G).

The degree of the permutation group G.

A set containing the defining generators for G.

Ngens(G) : GrpPerm -> RngIntElt

The number of defining generators for G.

The generic group containing G, i.e. the symmetric group in which G is naturally embedded.

The parent group G for the permutation g.

The natural G-set for the permutation group G.

• Consider the group G of order 648 generated by the permutations (1,6,7)(2,5,8,3,4,9)(11,12) and (1,3)(4,9,12)(5,8,10,6,7,11).

> G := PermutationGroup< 12 | (1,6,7)(2,5,8,3,4,9)(11,12),
>                             (1,3)(4,9,12)(5,8,10,6,7,11) >;
> G;
Permutation group G acting on a set of cardinality 12
    (1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
    (1, 3)(4, 9, 12)(5, 8, 10, 6, 7, 11)
> G.1;
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
> G.1*G.2;
(1, 7, 3, 9, 2, 8)(4, 12, 5, 10, 6, 11)
> Degree(G);
12
> GSet(G);
GSet{ 1 .. 12 }
> Generic(G);
Symmetric group acting on a set of cardinality 12
Order = 479001600 = 2^10 * 3^5 * 5^2 * 7 * 11
> Generators(G);
{ 
    (1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12),
    (1, 3)(4, 9, 12)(5, 8, 10, 6, 7, 11)
 }
> Ngens(G);
2
> x := G ! (1,6,7)(2,5,8,3,4,9)(11,12);
> x;
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
> Parent(x);
Permutation group G acting on a set of cardinality 12
Order = 648 = 2^3 * 3^4
    (1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
    (1, 3)(4, 9, 12)(5, 8, 10, 6, 7, 11)