If only the permutation image of a single element is needed, it might not be worth to create the action homomorphism, the following operations yield the permutation image and cycles of a single element.
Permutation( g, Omega[, gens, acts][, act] ) F
Permutation( g, xset ) F
computes the permutation that corresponds to the action of g on the
domain Omega or the UnderlyingDomain value of the external set
xset, respectively.
If g does not leave the domain invariant, or does not map injectively,
fail is returned.
PermutationCycle( g, Omega, pnt[, act] ) O
computes the permutation that represents the cycle of pnt under the action of the elemnt g
gap> Permutation((1,2,3)(4,5)(6,7),[4..7]); (1,2)(3,4) gap> PermutationCycle((1,2,3)(4,5)(6,7),[4..7],4); (1,2)
Cycle( g, Omega, pnt[, act] ) O
returns a list of the points in the cycle of pnt under the action of the element g.
CycleLength( g, Omega, pnt[, act] ) O
returns the length of the cycle of pnt under the action of the element g.
Cycles( g, Omega[, act] ) O
returns a list of the cycles (as lists of points) of the action of the element g.
CycleLengths( g, Omega, [, act] ) O
returns the lengths of all the cycles under the action of the element g on Omega.
gap> Cycle((1,2,3)(4,5)(6,7),[4..7],4); [ 4, 5 ] gap> CycleLength((1,2,3)(4,5)(6,7),[4..7],4); 2 gap> Cycles((1,2,3)(4,5)(6,7),[4..7]); [ [ 4, 5 ], [ 6, 7 ] ] gap> CycleLengths((1,2,3)(4,5)(6,7),[4..7]); [ 2, 2 ]
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GAP 4 manual