  
  [1X2 [33X[0;0YA sample calculation with [5XSophus[105X[101X[1X[133X[101X
  
  [33X[0;0YBefore  listing  the  functions of [5XSophus[105X we present a sample calculation to
  show  the  reader  what  [5XSophus[105X  is able to compute. We list the isomorphism
  types of the 7-dimensional nilpotent Lie algebras over [23X\mathbb F_2[123X.[133X
  
  [33X[0;0YThere is just one nilpotent Lie algebra with dimension 1 and dimension 2. We
  also  set  [3XL3[103X to be a list containing the abelian Lie algebra with dimension
  3.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL1 := [ AbelianLieAlgebra( GF(2), 1 ) ];;[127X[104X
    [4X[25Xgap>[125X [27XL2 := [ AbelianLieAlgebra( GF(2), 2 ) ];;[127X[104X
    [4X[25Xgap>[125X [27XL3 := [ AbelianLieAlgebra( GF(2), 3 ) ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAny   3-dimensional  non-abelian  nilpotent  Lie  algebra  is  an  immediate
  descendant  of  a  2-dimensional  Lie  algebra.  So  we  compute  the step-1
  descendants of [3XL1[1][103X and append them to [3XL3[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAppend( L3, Descendants( L2[1], 1 ));[127X[104X
    [4X[25Xgap>[125X [27XL3;[127X[104X
    [4X[28X[ <Lie algebra of dimension 3 over GF(2)>, [128X[104X
    [4X[28X  <Lie algebra of dimension 3 over GF(2)> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  the list of 4-dimensional Lie algebras. First we set [3XL4[103X to
  contain  the  4-dimensional  abelian Lie algebra. Then we compute the step-1
  descendants  of  the  3-dimensional algebras and append these descendants to
  [3XL4[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL4 := [ AbelianLieAlgebra( GF(2), 4 ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L3 do Append( L4, Descendants( i, 1 )); od;[127X[104X
    [4X[25Xgap>[125X [27XL4;[127X[104X
    [4X[28X[ <Lie algebra of dimension 4 over GF(2)>, [128X[104X
    [4X[28X  <Lie algebra of dimension 4 over GF(2)>, [128X[104X
    [4X[28X  <Lie algebra of dimension 4 over GF(2)> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe continue this way up to dimension~7.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL5 := [ AbelianLieAlgebra( GF(2), 5 ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L3 do Append( L5, Descendants( i, 2 )); od;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L4 do Append( L5, Descendants( i, 1 )); od;[127X[104X
    [4X[25Xgap>[125X [27XL6 := [ AbelianLieAlgebra( GF(2), 6 ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L3 do Append( L6, Descendants( i, 3 )); od;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L4 do Append( L6, Descendants( i, 2 )); od;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L5 do Append( L6, Descendants( i, 1 )); od;[127X[104X
    [4X[25Xgap>[125X [27XL7 := [ AbelianLieAlgebra( GF(2), 6 ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L4 do Append( L7, Descendants( i, 3 )); od;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L5 do Append( L7, Descendants( i, 2 )); od;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in L6 do Append( L7, Descendants( i, 1 )); od;[127X[104X
    [4X[25Xgap>[125X [27XLength( L7 );[127X[104X
    [4X[28X202[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  computation shows that there are 202 pairwise non-isomorphic nilpotent
  Lie algebras over [23X\mathbb F_2[123X.[133X
  
  [33X[0;0YLet  us  compute  the automorphism group of a nilpotent Lie algebra from our
  list.  We  compute  this  automorphism  group  in  the hybrid format used by
  [5XSophus[105X, then we compute this group as a standard [5XGAP[105X object.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAutomorphismGroupOfNilpotentLieAlgebra( L7[100] );[127X[104X
    [4X[28Xrec( agAutos := [ Aut: [ v.1, v.1+v.2, v.3, v.4, v.5, v.5+v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1, v.2+v.3, v.3, v.4, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1+v.3, v.2, v.3, v.4+v.5, v.5, v.6+v.7, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1+v.4, v.2, v.3+v.5, v.4+v.6, v.5+v.7, v.6+v.7, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1, v.2+v.4, v.3, v.4+v.5, v.5, v.6+v.7, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1+v.5, v.2, v.3, v.4+v.7, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1, v.2+v.5, v.3, v.4, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1+v.6, v.2, v.3, v.4+v.7, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1, v.2+v.6, v.3, v.4+v.7, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1+v.7, v.2, v.3, v.4, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1, v.2+v.7, v.3, v.4, v.5, v.6, v.7 ], [128X[104X
    [4X[28X      Aut: [ v.1, v.2, v.3+v.7, v.4, v.5, v.6, v.7 ] ], [128X[104X
    [4X[28X  agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], field := GF(2), [128X[104X
    [4X[28X  glAutos := [  ], glOper := [  ], glOrder := 1, [128X[104X
    [4X[28X  liealg := <Lie algebra of dimension 7 over GF(2)>, [128X[104X
    [4X[28X  one := Aut: [ v.1, v.2, v.3, v.4, v.5, v.6, v.7 ], prime := 2, size := 4096 [128X[104X
    [4X[28X )[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XAutomorphismGroup( L7[100] );[127X[104X
    [4X[28X<group with 12 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally let us check that two Lie algebras from our list are not isomorphic.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAreIsomorphicNilpotentLieAlgebras( L7[100], L7[100] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XAreIsomorphicNilpotentLieAlgebras( L7[100], L7[101] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
