In this section we outline two example computations with the functions of the previous chapter.
gap> G := MatExamples(2); <matrix group with 6 generators> gap> mats := GeneratorsOfGroup( G ); [ [ [ 2, -1, -1, 3 ], [ -1, 0, 1, -4 ], [ 0, 1, 2, -2 ], [ -1, 1, 2, -4 ] ], [ [ 1, 1, 1, -2 ], [ 1, 1, -1, 1 ], [ -2, 1, 0, -3 ], [ -1, 1, 0, -2 ] ], [ [ 1, 0, -1, 0 ], [ -2, 5, 5, -11 ], [ -1, 0, 2, 0 ], [ -1, 1, 2, -2 ] ], [ [ 0, 3, 2, -7 ], [ -1, 6, 4, -13 ], [ -1, 2, 1, -5 ], [ -1, 3, 2, -7 ] ], [ [ 0, 4, 5, -12 ], [ 2, -4, 1, 5 ], [ 1, -4, -1, 8 ], [ 1, -3, 0, 5 ] ], [ [ 1, 0, -1, 0 ], [ -2, 5, 5, -11 ], [ -1, 0, 2, 0 ], [ -1, 1, 2, -2 ] ] ] # calculate an isomorphism from G to a pcp-group gap> nat := IsomorphismPcpGroup( G );; gap> H := Image( nat ); Pcp-group with orders [ 2, 2, 3, 3, 4, 0, 0, 0 ] gap> h := GeneratorsOfGroup( H ); [ g1, g2, g3, g4, g5, g6, g7, g8 ] gap> mats2 := List( h, x -> PreImage( nat, x ) );; # take a random element of G gap> exp := [ 1, 1, 1, 1, 0, 0, 0, 0 ];; gap> g := MappedVector( exp, mats2 ); [ [ 229793843, -345584045, -503782245, 823202280 ], [ 397912065, -598518263, -872506665, 1425593295 ], [ 141954212, -213549855, -311309508, 508615375 ], [ 189806315, -285510521, -416211500, 680033928 ] ] # map g into the image of nat gap> i := ImageElm( nat, g ); g1*g2*g3*g4 # exponent vector gap> Exponents( i ); [ 1, 1, 1, 1, 0, 0, 0, 0 ] # compare the preimage with g gap> PreImagesRepresentative( nat, i ); [ [ 229793843, -345584045, -503782245, 823202280 ], [ 397912065, -598518263, -872506665, 1425593295 ], [ 141954212, -213549855, -311309508, 508615375 ], [ 189806315, -285510521, -416211500, 680033928 ] ] gap> last = g; true
gap> gens := [ [ [ 1746/1405, 524/7025, 418/1405, -77/2810 ], [ 815/843, 899/843, -1675/843, 415/281 ], [ -3358/4215, -3512/21075, 4631/4215, -629/1405 ], [ 258/1405, 792/7025, 1404/1405, 832/1405 ] ], [ [ -2389/2810, 3664/21075, 8942/4215, -35851/16860 ], [ 395/281, 2498/2529, -5105/5058, 3260/2529 ], [ 3539/2810, -13832/63225, -12001/12645, 87053/50580 ], [ 5359/1405, -3128/21075, -13984/4215, 40561/8430 ] ] ]; gap> H := Group( gens ); <matrix group with 2 generators> gap> RadicalSeriesSolvableMatGroup( H ); [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 0, 1, -197/414 ], [ 1, 0, -3, 2 ], [ 0, 1, 55/4, -55/8 ] ], [ [ 0, 1, 55/4, -55/8 ], [ 1, 4/15, 2/3, 1/6 ] ], [ [ 1, 4/15, 2/3, 1/6 ] ], [ ] ]
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