PermGroupRepresentation( G, limit )
PermGroupRepresentation tries to find a permutation representation of
G of degree at most limit. The function either returns a permutation
group or false if no such representation was found.
Note that false does not imply that no such permutation
representation exists. If a permutation representation for G is
already known it will be returned independent of its degree.
The function tries to find a set of vectors of size at most limit closed under the operation of G such that the set spans the whole vector space; it implements parts of the base-point selection algorithm described in [10].
gap> m1 := [[0,1],[1,0]] * Z(9);;
gap> m2 := [[1,1],[1,0]] * Z(9);;
gap> G := Group( m1, m2 );;
gap> P := PermGroupRepresentation( G, 100 );
Group( ( 1,15, 4,21, 2,24, 7,30)( 3,18, 5,12, 6,27, 8, 9)
(10,16,19,22,14,26,29,32)(11,25,20,31,13,17,28,23),
( 1,15,19,31)( 2,24,29,23)( 3,18,22,11)( 4,21,14,17)( 5,12,26,20)
( 6,27,32,13)( 7,30,10,25)( 8, 9,16,28) )
# note that <limit> is ignored if a representation is known
gap> P := PermGroupRepresentation( G, 2 );
Group( ( 1,15, 4,21, 2,24, 7,30)( 3,18, 5,12, 6,27, 8, 9)
(10,16,19,22,14,26,29,32)(11,25,20,31,13,17,28,23),
( 1,15,19,31)( 2,24,29,23)( 3,18,22,11)( 4,21,14,17)( 5,12,26,20)
( 6,27,32,13)( 7,30,10,25)( 8, 9,16,28) )
OrbitMat( G, vec, base, limit )
OrbitMat computes the orbit of vec under the operation of G. The
function will return false if this orbit is larger then limit.
Otherwise the orbit is return as list of vectors and base, which must
be supplied as an empty list, now contains a list of basis vectors
spanning the vector space generated by the orbit.
GAP 3.4.4