#We find all primitive subgroups of $GL(16,5)$ with parameters $e=16$, $a=1$ #In the notations of Theorem 5.1, if $H$ is a primitive solvable subgroup, then it possesses a series #$$1 G:=GL(16,5);; gap> z:=Z(5);; gap> v:=[z,z^2,z^3,0*z,z^2,z^3,z,z^3,0*z,z,z^2,z,z^3,z^2,z,z^3];; gap> result:=[];; #Now we run the following loop, where gens is the list of generators from generators-GL(16,5).txt. Each entry of the list result has three items: the first is the #number of the group in the list gens, the second is the order of the group, and the third is the order of the stabilizer of v in H. #If the order of the stabilizer is 1, then |v^H=|H| and Corollary 2.5 can be applied. gap> for j in [1..Size(gens)] do > H:=Subgroup(G,gens[j]);; > n:=Size(H); > m:=Size(v^H); > result[j]:=[j,n,n/m]; > od; gap> result; [ [ 1, 17408, 1 ], [ 2, 9216, 1 ], [ 3, 34816, 1 ], [ 4, 18432, 1 ], [ 5, 69632, 1 ], [ 6, 139264, 1 ], [ 7, 17408, 1 ], [ 8, 25600, 1 ], [ 9, 34816, 1 ], [ 10, 30720, 1 ], [ 11, 30720, 1 ], [ 12, 51200, 1 ], [ 13, 51200, 1 ], [ 14, 51200, 1 ], [ 15, 51200, 1 ], [ 16, 69632, 1 ], [ 17, 61440, 1 ], [ 18, 61440, 1 ], [ 19, 102400, 1 ], [ 20, 102400, 1 ], [ 21, 102400, 1 ], [ 22, 102400, 1 ], [ 23, 102400, 1 ], [ 24, 102400, 1 ], [ 25, 102400, 1 ], [ 26, 139264, 1 ], [ 27, 122880, 1 ], [ 28, 204800, 1 ], [ 29, 204800, 1 ], [ 30, 204800, 2 ], [ 31, 204800, 1 ], [ 32, 204800, 1 ], [ 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7962624, 1 ], [ 1080, 7962624, 1 ], [ 1081, 10616832, 1 ], [ 1082, 15925248, 2 ], [ 1083, 15925248, 1 ], [ 1084, 15925248, 1 ], [ 1085, 31850496, 2 ] ] # All the groups H for which |v^H|<> |H| are computed to the list res1 gap> res1:=Filtered(result,x->x[3]<>1);; gap> Size(res1); 44 #We choose another vector v and repeat the cycle for the remaining groups gap> v:=[z,z^2,z^3,0*z,z^2,z^3,z,z^3,0*z,z,0*z^2,0*z,0*z^3,0*z^2,0*z,0*z^3];; gap> result1:=[]; [ ] gap> for j in res1 do > H:=Subgroup(GL(16,5),gens[j[1]]); > m:=Size(v^H); > Append(result1,[[j[1],j[2],j[2]/m]]); > Display([j[1],j[2],j[2]/m]); > od; [ 30, 204800, 1 ] [ 34, 204800, 1 ] [ 35, 409600, 1 ] [ 455, 110592, 1 ] [ 470, 165888, 1 ] [ 485, 165888, 1 ] [ 512, 331776, 1 ] [ 526, 331776, 1 ] [ 527, 331776, 1 ] [ 534, 331776, 1 ] [ 545, 331776, 1 ] [ 561, 497664, 1 ] [ 569, 663552, 1 ] [ 572, 663552, 1 ] [ 574, 663552, 1 ] [ 581, 663552, 1 ] [ 582, 663552, 1 ] [ 587, 663552, 1 ] [ 607, 663552, 1 ] [ 608, 663552, 1 ] [ 615, 663552, 1 ] [ 628, 995328, 1 ] [ 630, 1327104, 1 ] [ 631, 1327104, 1 ] [ 632, 1327104, 1 ] [ 633, 1327104, 1 ] [ 638, 1327104, 1 ] [ 641, 1327104, 1 ] [ 642, 1327104, 1 ] [ 659, 1990656, 1 ] [ 660, 1990656, 1 ] [ 662, 2654208, 1 ] [ 664, 3981312, 1 ] [ 666, 3981312, 1 ] [ 669, 3981312, 1 ] [ 673, 3981312, 1 ] [ 674, 7962624, 1 ] [ 773, 61440, 1 ] [ 925, 663552, 1 ] [ 1026, 1327104, 2 ] [ 1036, 1990656, 1 ] [ 1070, 3981312, 2 ] [ 1082, 15925248, 2 ] [ 1085, 31850496, 2 ] #Now we choose groups such that |v^H|<>|H| in the list res2 gap> res2:=Filtered(result1,x->x[3]<>1); #We choose another vector v and repeat the cycle for the remaining groups gap> v:=[0*z,0*z^2,z,z,0*z,z,z,z,z,z,z,z^2,z,0*z^2,z,0*z^2];; gap> result2:=[]; gap> for j in res2 do > H:=Subgroup(GL(16,5),gens[j[1]]);; > n:=j[2];; > m:=Size(v^H);; > Append(result2,[[j[1],j[2],n/m]]); > Display([j[1],j[2],n/m]); > od; [ 1026, 1327104, 1 ] [ 1070, 3981312, 1 ] [ 1082, 15925248, 1 ] [ 1085, 31850496, 1 ]