GAP Manual: 8.2. Cycle

Cycle( g, d )
Cycle( g, d, operation )

Cycle returns the orbit of the point d, which may be an object of arbitrary type, under the group element g as a list of points.

The points e in the cycle of d under the group element g are those for which a power g^i exists such that d^(g^i) = e.

The first point in the list returned by Cycle is the point d itself, the ordering of the other points is such that each point is the image of the previous point.

Cycle accepts a function operation of two arguments d and g as optional third argument, which specifies how the element g operates (see Other Operations).

    gap> Cycle( (1,5,3,8)(4,6,7), 3 );
    [ 3, 8, 1, 5 ]
    gap> Cycle( (1,5,3,8)(4,6,7), [3,4], OnPairs );
    [ [ 3, 4 ], [ 8, 6 ], [ 1, 7 ], [ 5, 4 ], [ 3, 6 ], [ 8, 7 ],
      [ 1, 4 ], [ 5, 6 ], [ 3, 7 ], [ 8, 4 ], [ 1, 6 ], [ 5, 7 ] ] 

Cycle calls
Domain([g]).operations.Cycle( g, d, operation )
and returns the value. Note that the third argument is not optional for the functions called this way.

The default function called this way is GroupElementsOps.Cycle, which starts with d and applies g to the last point repeatedly until d is reached again. Special categories of group elements overlay this default function with more efficient functions.