gap> uu:=(1,3)(2,4);ff:=(1,8)(4,5);dd:=(5,7)(6,8); (1,3)(2,4) (1,8)(4,5) (5,7)(6,8) gap> bb:=(3,6)(2,7);rr:=(2,5)(1,6);ll:=(4,7)(3,8); (2,7)(3,6) (1,6)(2,5) (3,8)(4,7) gap> H:=Group(uu,ff,rr,dd,bb,ll); Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8) (4,7) ) gap> gap> Lattice(H); LatticeSubgroups( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ) ) gap> L:=Lattice(H); LatticeSubgroups( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ) ) gap> L[1]; Error, List Element: must be a list gap> NormalSubgroups(H); [ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,8)(3,6) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (1,3)(6,8), (1,6)(2,5)(3,8)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6) (2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (3,6,8)(4,5,7), (1,6,8)(2,5,7), (1,3,6)(4,7,5) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (5,7)(6,8), (1,6)(4,7), (1,8)(2,7), (2,5)(3,8) ] ) ] gap> N:=NormalSubgroups(H); [ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,8)(3,6) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (1,3)(6,8), (1,6)(2,5)(3,8)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6) (2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (3,6,8)(4,5,7), (1,6,8)(2,5,7), (1,3,6)(4,7,5) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (5,7)(6,8), (1,6)(4,7), (1,8)(2,7), (2,5)(3,8) ] ) ] gap> Size(N); Error, must be a domain or a set in Size( N ) called from main loop brk> quit; gap> Length(N); 7 gap> for i from 1 to 7 do Size(N[i]); od; Syntax error: in expected for i from 1 to 7 do Size(N[i]); od; ^ gap> for i to 7 do Size(N[i]); od; Syntax error: in expected for i to 7 do Size(N[i]); od; ^ gap> for i in [1..7] do Size(N[i]); od; gap> for i in [1..7] do Size(N[i]) od; Syntax error: ; expected for i in [1..7] do Size(N[i]) od; ^ gap> for i in [1..7] do Print(Size(N[i])); od; 1444164896gap> gap> for i in [1..7] do Print(Size(N[i]),` `); od; Syntax error: expression expected for i in [1..7] do Print(Size(N[i]),` `); od; ^ gap> for i in [1..7] do Print(Size(N[i])," "); od; 1 4 4 4 16 48 96 gap> gap> N[6]; Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (3,6,8)(4,5,7), (1,6,8)(2,5,7), (1,3,6)(4,7,5) ] ) gap> N[2]; Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7) ] ) gap> IsCyclic(N[2]); false gap> IsCyclic(N[3]); false gap> IsCyclic(N[4]); false gap> IsAbelian(N[2]); true gap> IsAbelian(N[3]); true gap> IsAbelian(N[4]); true gap> IsElementaryAbelian(N[2]); true gap> IsElementaryAbelian(N[3]); true gap> IsElementaryAbelian(N[4]); true gap> IsElementaryAbelian(N[5]); true gap> for i in [1..7] do Print(Size(N[i])," "); od; 1 4 4 4 16 48 96 gap> gap> N6:=NormalSubgroups(N[6]); [ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,8)(3,6) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (1,3)(2,5)(4,7)(6,8), (1,8)(2,4)(3,6)(5,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (1,3)(2,7)(4,5)(6,8), (1,8)(2,5)(3,6)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (1,3)(6,8), (1,8)(2,4)(3,6)(5,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6) (2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (3,6,8)(4,5,7), (1,3,8)(2,4,7), (1,8,3)(2,4,5) ] ) ] gap> Length(N6); 8 gap> IsAbelian(N[6]); false gap> for i in [1..8] do Print(Size(N6[i])," "); od; 1 4 4 4 4 4 16 48 gap> gap> IsElementaryAbelian(N6[2]); true gap> IsElementaryAbelian(N6[3]); true gap> IsElementaryAbelian(N6[4]); true gap> IsElementaryAbelian(N6[5]); true gap> IsElementaryAbelian(N6[6]); true gap> IsElementaryAbelian(N6[7]); true gap> IsElementaryAbelian(N6[8]); false gap> MaximalSubgroups(H); [ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (3,6,8)(4,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,7)(4,5), (1,6,8)(2,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,5)(4,7), (2,4)(5,7), (1,3,6)(4,7,5), (1,6)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,7)(4,5), (2,5)(4,7), (1,8,3)(2,7,4), (1,8)(2,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (3,6,8)(4,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (2,7,5)(3,6,8), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (2,5,4)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (2,4,7)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (3,6,8)(4,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (2,7,5)(3,6,8), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8) (4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (2,5,4)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (2,4,7)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8) (4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (5,7)(6,8), (2,4)(6,8), (2,5)(4,7), (1,3)(6,8), (1,6)(3,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (3,6)(4,5), (2,7)(3,6), (2,4)(5,7), (1,8)(3,6), (1,3)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (3,8)(4,7), (2,5)(3,8), (2,4)(5,7), (1,6)(3,8), (1,3)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (1,3)(6,8), (1,6)(3,8), (3,6,8)(4,5,7) ] ) ] gap> M:=MaximalSubgroups(H); [ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (3,6,8)(4,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,7)(4,5), (1,6,8)(2,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (2,5)(4,7), (2,4)(5,7), (1,3,6)(4,7,5), (1,6)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,7)(4,5), (2,5)(4,7), (1,8,3)(2,7,4), (1,8)(2,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (3,6,8)(4,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (2,7,5)(3,6,8), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (2,5,4)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(6,8), (1,6)(3,8), (2,4,7)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (3,6,8)(4,5,7), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (2,7,5)(3,6,8), (5,7)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8) (4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (2,5,4)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7), (2,4,7)(3,6,8), (2,4)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8) (4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (5,7)(6,8), (2,4)(6,8), (2,5)(4,7), (1,3)(6,8), (1,6)(3,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (3,6)(4,5), (2,7)(3,6), (2,4)(5,7), (1,8)(3,6), (1,3)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), [ (3,8)(4,7), (2,5)(3,8), (2,4)(5,7), (1,6)(3,8), (1,3)(6,8) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7) (3,6), (3,8)(4,7) ), [ (2,4)(5,7), (2,5)(4,7), (1,3)(6,8), (1,6)(3,8), (3,6,8)(4,5,7) ] ) ] gap> Length(M); 16 gap> for i in [1..16] do Print(Size(M[i])," "); od; 24 24 24 24 24 24 24 24 24 24 24 24 32 32 32 48 gap> gap> quit;