  
  [1X49 [33X[0;0YGroup Products[133X[101X
  
  [33X[0;0YThis  chapter  describes  the  various  group product constructions that are
  possible in [5XGAP[105X.[133X
  
  [33X[0;0YAt  the  moment  for some of the products methods are available only if both
  factors  are  given  in the same representation or only for certain types of
  groups  such  as  permutation  groups  and pc groups when the product can be
  naturally represented as a group of the same kind.[133X
  
  [33X[0;0Y[5XGAP[105X  does not guarantee that a product of two groups will be in a particular
  representation.  (Exceptions are [2XWreathProductImprimitiveAction[102X ([14X49.4-2[114X) and
  [2XWreathProductProductAction[102X  ([14X49.4-3[114X) which are construction that makes sense
  only for permutation groups, see [2XWreathProduct[102X ([14X49.4-1[114X)).[133X
  
  [33X[0;0Y[5XGAP[105X  however  will try to choose an efficient representation, so products of
  permutation  groups or pc groups often will be represented as a group of the
  same kind again.[133X
  
  [33X[0;0YTherefore  the only guaranteed way to relate a product to its factors is via
  the embedding and projection homomorphisms, see [14X49.6[114X.[133X
  
  
  [1X49.1 [33X[0;0YDirect Products[133X[101X
  
  [33X[0;0YThe  direct  product  of  groups  is  the  cartesian  product  of the groups
  (considered as element sets) with component-wise multiplication.[133X
  
  [1X49.1-1 DirectProduct[101X
  
  [33X[1;0Y[29X[2XDirectProduct[102X( [3XG[103X[, [3XH[103X, [3X...[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XDirectProductOp[102X( [3Xlist[103X, [3Xexpl[103X ) [32X operation[133X
  
  [33X[0;0YThese  functions  construct  the  direct  product  of  the  groups  given as
  arguments. [2XDirectProduct[102X takes an arbitrary positive number of arguments and
  calls  the  operation  [2XDirectProductOp[102X,  which  takes exactly two arguments,
  namely  a  nonempty list [3Xlist[103X of groups and one of these groups, [3Xexpl[103X. (This
  somewhat  strange  syntax allows the method selection to choose a reasonable
  method  for  special cases, e.g., if all groups are permutation groups or pc
  groups.)[133X
  
  [33X[0;0Y[5XGAP[105X  will  try to choose an efficient representation for the direct product.
  For  example  the direct product of permutation groups will be a permutation
  group again and the direct product of pc groups will be a pc group.[133X
  
  [33X[0;0YIf the groups are in different representations a generic direct product will
  be  formed  which  may  not be particularly efficient for many calculations.
  Instead  it  may  be worth to convert all factors to a common representation
  first, before forming the product.[133X
  
  [33X[0;0YFor  a direct product [22XP[122X, calling [2XEmbedding[102X ([14X32.2-11[114X) with [22XP[122X and [22Xn[122X yields the
  homomorphism  embedding the [22Xn[122X-th factor into [22XP[122X; calling [2XProjection[102X ([14X32.2-12[114X)
  with [3XP[103X and [3Xn[103X yields the projection of [22XP[122X onto the [22Xn[122X-th factor, see [14X49.6[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3),(1,2));;[127X[104X
    [4X[25Xgap>[125X [27Xd:=DirectProduct(g,g,g);[127X[104X
    [4X[28XGroup([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ])[128X[104X
    [4X[25Xgap>[125X [27XSize(d);[127X[104X
    [4X[28X216[128X[104X
    [4X[25Xgap>[125X [27Xe:=Embedding(d,2);[127X[104X
    [4X[28X2nd embedding into Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), [128X[104X
    [4X[28X  (7,8) ])[128X[104X
    [4X[25Xgap>[125X [27XImage(e,(1,2));[127X[104X
    [4X[28X(4,5)[128X[104X
    [4X[25Xgap>[125X [27XImage(Projection(d,2),(1,2,3)(4,5)(8,9));[127X[104X
    [4X[28X(1,2)[128X[104X
    [4X[25Xgap>[125X [27Xf:=FreeGroup("a","b");;[127X[104X
    [4X[25Xgap>[125X [27Xg:=f/ParseRelators(f,"a2,b3,(ab)5");[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27Xf2:=FreeGroup("x","y");;[127X[104X
    [4X[25Xgap>[125X [27Xh:=f2/ParseRelators(f2,"x2,y4,xy=Yx");[127X[104X
    [4X[28X<fp group on the generators [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27Xd:=DirectProduct(g,h);[127X[104X
    [4X[28X<fp group on the generators [ a, b, x, y ]>[128X[104X
  [4X[32X[104X
  
  
  [1X49.2 [33X[0;0YSemidirect Products[133X[101X
  
  [33X[0;0YThe  semidirect  product  of  a  group  [22XN[122X  with  a group [22XG[122X acting on [22XN[122X via a
  homomorphism  [22Xα[122X  from  [22XG[122X  into  the automorphism group of [22XN[122X is the cartesian
  product [22XG × N[122X with the multiplication [22X(g, n) ⋅ (h, m) = (gh, n^{h^α}m)[122X.[133X
  
  
  [1X49.2-1 [33X[0;0YSemidirectProduct[133X[101X
  
  [33X[1;0Y[29X[2XSemidirectProduct[102X( [3XG[103X, [3Xalpha[103X, [3XN[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSemidirectProduct[102X( [3Xautgp[103X, [3XN[103X ) [32X operation[133X
  
  [33X[0;0Yconstructs  the  semidirect product of [3XN[103X with [3XG[103X acting via [3Xalpha[103X, which must
  be a homomorphism from [3XG[103X into a group of automorphisms of [3XN[103X.[133X
  
  [33X[0;0YIf  [3XN[103X  is  a  group,  [3Xalpha[103X  must  be  a homomorphism from [3XG[103X into a group of
  automorphisms of [3XN[103X.[133X
  
  [33X[0;0YIf [3XN[103X is a full row space over a field [3XF[103X, [3Xalpha[103X must be a homomorphism from [3XG[103X
  into  a  matrix group of the right dimension over a subfield of [3XF[103X, or into a
  permutation group (in this case permutation matrices are taken).[133X
  
  [33X[0;0YIn  the  second variant, [3Xautgp[103X must be a group of automorphism of [3XN[103X, it is a
  shorthand  for  [10XSemidirectProduct([3Xautgp[103X[10X,IdentityMapping([3Xautgp[103X[10X),[3XN[103X[10X)[110X. Note that
  (unless   [3Xautgrp[103X  has  been  obtained  by  the  operation  [2XAutomorphismGroup[102X
  ([14X40.7-1[114X))  you  have  to  test [2XIsGroupOfAutomorphisms[102X ([14X40.7-2[114X) for [3Xautgrp[103X to
  ensure that [5XGAP[105X knows that [3Xautgrp[103X consists of group automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:=AbelianGroup(IsPcGroup,[5,5]);[127X[104X
    [4X[28X<pc group of size 25 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xau:=DerivedSubgroup(AutomorphismGroup(n));;[127X[104X
    [4X[25Xgap>[125X [27XSize(au);[127X[104X
    [4X[28X120[128X[104X
    [4X[25Xgap>[125X [27Xp:=SemidirectProduct(au,n);;[127X[104X
    [4X[25Xgap>[125X [27XSize(p);[127X[104X
    [4X[28X3000[128X[104X
    [4X[25Xgap>[125X [27Xn:=Group((1,2),(3,4));;[127X[104X
    [4X[25Xgap>[125X [27Xau:=AutomorphismGroup(n);;[127X[104X
    [4X[25Xgap>[125X [27Xau:=First(AsSet(au),i->Order(i)=3);;[127X[104X
    [4X[25Xgap>[125X [27Xau:=Group(au);[127X[104X
    [4X[28X<group with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XIsGroupOfAutomorphisms(au);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSemidirectProduct(au,n);[127X[104X
    [4X[28X<pc group with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xn:=AbelianGroup(IsPcGroup,[2,2]);[127X[104X
    [4X[28X<pc group of size 4 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xau:=AutomorphismGroup(n);;[127X[104X
    [4X[25Xgap>[125X [27Xapc:=IsomorphismPcGroup(au);;[127X[104X
    [4X[25Xgap>[125X [27Xg:=Image(apc);[127X[104X
    [4X[28XGroup([ f1, f2 ])[128X[104X
    [4X[25Xgap>[125X [27Xapci:=InverseGeneralMapping(apc);;[127X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism(apci);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xp:=SemidirectProduct(g,apci,n);[127X[104X
    [4X[28X<pc group of size 24 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismGroups(p,Group((1,2,3,4),(1,2))) <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSemidirectProduct(SU(3,3),GF(9)^3);[127X[104X
    [4X[28X<matrix group of size 4408992 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSemidirectProduct(Group((1,2,3),(2,3,4)),GF(5)^4);[127X[104X
    [4X[28X<matrix group of size 7500 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xg:=Group((3,4,5),(1,2,3));;[127X[104X
    [4X[25Xgap>[125X [27Xmats:=[[[Z(2^2),0*Z(2)],[0*Z(2),Z(2^2)^2]],[127X[104X
    [4X[25X>[125X [27X         [[Z(2)^0,Z(2)^0], [Z(2)^0,0*Z(2)]]];;[127X[104X
    [4X[25Xgap>[125X [27Xhom:=GroupHomomorphismByImages(g,Group(mats),[g.1,g.2],mats);;[127X[104X
    [4X[25Xgap>[125X [27XSemidirectProduct(g,hom,GF(4)^2);[127X[104X
    [4X[28X<matrix group of size 960 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSemidirectProduct(g,hom,GF(16)^2);[127X[104X
    [4X[28X<matrix group of size 15360 with 4 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  a  semidirect product [22XP[122X of [3XG[103X with [3XN[103X, calling [2XEmbedding[102X ([14X32.2-11[114X) with [22XP[122X
  and  [10X1[110X  yields  the embedding of [3XG[103X, calling [2XEmbedding[102X ([14X32.2-11[114X) with [22XP[122X and [10X2[110X
  yields  the  embedding  of [3XN[103X; calling [2XProjection[102X ([14X32.2-12[114X) with [3XP[103X yields the
  projection of [22XP[122X onto [3XG[103X, see [14X49.6[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize(Image(Embedding(p,1)));[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XEmbedding(p,2);[127X[104X
    [4X[28X[ f1, f2 ] -> [ f3, f4 ][128X[104X
    [4X[25Xgap>[125X [27XProjection(p);[127X[104X
    [4X[28X[ f1, f2, f3, f4 ] -> [ f1, f2, <identity> of ..., <identity> of ... ][128X[104X
  [4X[32X[104X
  
  
  [1X49.3 [33X[0;0YSubdirect Products[133X[101X
  
  [33X[0;0YThe subdirect product of the groups [22XG[122X and [22XH[122X with respect to the epimorphisms
  [22Xφ:  G  → A[122X and [22Xψ: H → A[122X (for a common group [22XA[122X) is the subgroup of the direct
  product  [22XG  ×  H[122X consisting of the elements [22X(g,h)[122X for which [22Xg^φ = h^ψ[122X. It is
  the pull-back of the following diagram.[133X
  
                     G
                     | phi
               psi   V
            H  --->  A
  
  [1X49.3-1 SubdirectProduct[101X
  
  [33X[1;0Y[29X[2XSubdirectProduct[102X( [3XG[103X, [3XH[103X, [3XGhom[103X, [3XHhom[103X ) [32X function[133X
  
  [33X[0;0Yconstructs the subdirect product of [3XG[103X and [3XH[103X with respect to the epimorphisms
  [3XGhom[103X from [3XG[103X onto a group [22XA[122X and [3XHhom[103X from [3XH[103X onto the same group [22XA[122X.[133X
  
  [33X[0;0YFor  a subdirect product [22XP[122X, calling [2XProjection[102X ([14X32.2-12[114X) with [22XP[122X and [22Xn[122X yields
  the projection on the [22Xn[122X-th factor. (In general the factors do not embed into
  a subdirect product.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3),(1,2));[127X[104X
    [4X[28XGroup([ (1,2,3), (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27Xhom:=GroupHomomorphismByImagesNC(g,g,[(1,2,3),(1,2)],[(),(1,2)]);[127X[104X
    [4X[28X[ (1,2,3), (1,2) ] -> [ (), (1,2) ][128X[104X
    [4X[25Xgap>[125X [27Xs:=SubdirectProduct(g,g,hom,hom);[127X[104X
    [4X[28XGroup([ (1,2,3), (1,2)(4,5), (4,5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XSize(s);[127X[104X
    [4X[28X18[128X[104X
    [4X[25Xgap>[125X [27Xp:=Projection(s,2);[127X[104X
    [4X[28X2nd projection of Group([ (1,2,3), (1,2)(4,5), (4,5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XImage(p,(1,3,2)(4,5,6));[127X[104X
    [4X[28X(1,2,3)[128X[104X
  [4X[32X[104X
  
  [1X49.3-2 SubdirectProducts[101X
  
  [33X[1;0Y[29X[2XSubdirectProducts[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0Ythis  function computes all subdirect products of [3XG[103X and [3XH[103X up to conjugacy in
  the  direct  product  of Parent([3XG[103X) and Parent([3XH[103X). The subdirect products are
  returned as subgroups of this direct product.[133X
  
  
  [1X49.4 [33X[0;0YWreath Products[133X[101X
  
  [33X[0;0YThe  wreath  product  of  a  group  [22XG[122X with a permutation group [22XP[122X acting on [22Xn[122X
  points is the semidirect product of the normal subgroup [22X[3XG[103X^n[122X with the group [22XP[122X
  which acts on [22X[3XG[103X^n[122X by permuting the components.[133X
  
  [33X[0;0YNote  that  [5XGAP[105X always considers the domain of a permutation group to be the
  points  moved  by elements of the group as returned by [2XMovedPoints[102X ([14X42.3-3[114X),
  i.e. it is not possible to have a domain to include fixed points, I.e. [22XP = ⟨
  (1,2,3)  ⟩[122X  and  [22XP  =  ⟨ (1,3,5) ⟩[122X result in isomorphic wreath products. (If
  fixed  points  are  desired the wreath product [22XG ≀ T[122X has to be formed with a
  transitive overgroup [22XT[122X of [22XP[122X and then the pre-image of [22XP[122X under the projection
  [22XG ≀ T → T[122X has to be taken.)[133X
  
  [1X49.4-1 WreathProduct[101X
  
  [33X[1;0Y[29X[2XWreathProduct[102X( [3XG[103X, [3XH[103X[, [3Xhom[103X] ) [32X operation[133X
  [33X[1;0Y[29X[2XStandardWreathProduct[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  
  [33X[0;0Y[10XWreathProduct[110X constructs the wreath product of the group [3XG[103X with the group [3XH[103X,
  acting as a permutation group.[133X
  
  [33X[0;0YIf  a  third  argument [3Xhom[103X is given, it must be a homomorphism from [3XH[103X into a
  permutation  group,  and  the  action  of  this group on its moved points is
  considered.[133X
  
  [33X[0;0YIf only two arguments are given, [3XH[103X must be a permutation group.[133X
  
  [33X[0;0Y[10XStandardWreathProduct[110X  returns  the  wreath  product for the (right regular)
  permutation action of [3XH[103X on its elements.[133X
  
  [33X[0;0YFor a wreath product [22XW[122X of [3XG[103X with a permutation group [22XP[122X of degree [22Xn[122X and [22X1 ≤ i
  ≤  n[122X  calling  [2XEmbedding[102X ([14X32.2-11[114X) with [22XW[122X and [22Xi[122X yields the embedding of [3XG[103X in
  the [22Xi[122X-th component of the direct product of the base group [22X[3XG[103X^n[122X of [22XW[122X. For [22Xi =
  n+1[122X,  [2XEmbedding[102X  ([14X32.2-11[114X)  yields  the  embedding  of  [22XP[122X  into  [22XW[122X.  Calling
  [2XProjection[102X  ([14X32.2-12[114X)  with [22XW[122X yields the projection onto the acting group [22XP[122X,
  see [14X49.6[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3),(1,2));[127X[104X
    [4X[28XGroup([ (1,2,3), (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27Xp:=Group((1,2,3));[127X[104X
    [4X[28XGroup([ (1,2,3) ])[128X[104X
    [4X[25Xgap>[125X [27Xw:=WreathProduct(g,p);[127X[104X
    [4X[28XGroup([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8), [128X[104X
    [4X[28X  (1,4,7)(2,5,8)(3,6,9) ])[128X[104X
    [4X[25Xgap>[125X [27XSize(w);[127X[104X
    [4X[28X648[128X[104X
    [4X[25Xgap>[125X [27XEmbedding(w,1);[127X[104X
    [4X[28X1st embedding into Group( [ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), [128X[104X
    [4X[28X  (7,8), (1,4,7)(2,5,8)(3,6,9) ] )[128X[104X
    [4X[25Xgap>[125X [27XImage(Embedding(w,3));[127X[104X
    [4X[28XGroup([ (7,8,9), (7,8) ])[128X[104X
    [4X[25Xgap>[125X [27XImage(Embedding(w,4));[127X[104X
    [4X[28XGroup([ (1,4,7)(2,5,8)(3,6,9) ])[128X[104X
    [4X[25Xgap>[125X [27XImage(Projection(w),(1,4,8,2,6,7,3,5,9));[127X[104X
    [4X[28X(1,2,3)[128X[104X
  [4X[32X[104X
  
  [1X49.4-2 WreathProductImprimitiveAction[101X
  
  [33X[1;0Y[29X[2XWreathProductImprimitiveAction[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YFor  two  permutation  groups  [3XG[103X  and [3XH[103X, this function constructs the wreath
  product of [3XG[103X and [3XH[103X in the imprimitive action. If [3XG[103X acts on [22Xl[122X points and [3XH[103X on
  [22Xm[122X  points this action will be on [22Xl ⋅ m[122X points, it will be imprimitive with [22Xm[122X
  blocks of size [22Xl[122X each.[133X
  
  [33X[0;0YThe  operations [2XEmbedding[102X ([14X32.2-11[114X) and [2XProjection[102X ([14X32.2-12[114X) operate on this
  product as described for general wreath products.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xw:=WreathProductImprimitiveAction(g,p);;[127X[104X
    [4X[25Xgap>[125X [27XLargestMovedPoint(w);[127X[104X
    [4X[28X9[128X[104X
  [4X[32X[104X
  
  [1X49.4-3 WreathProductProductAction[101X
  
  [33X[1;0Y[29X[2XWreathProductProductAction[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YFor  two  permutation  groups  [3XG[103X  and [3XH[103X, this function constructs the wreath
  product  in  product  action.  If  [3XG[103X acts on [22Xl[122X points and [3XH[103X on [22Xm[122X points this
  action will be on [22Xl^m[122X points.[133X
  
  [33X[0;0YThe  operations [2XEmbedding[102X ([14X32.2-11[114X) and [2XProjection[102X ([14X32.2-12[114X) operate on this
  product as described for general wreath products.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xw:=WreathProductProductAction(g,p);[127X[104X
    [4X[28X<permutation group of size 648 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XLargestMovedPoint(w);[127X[104X
    [4X[28X27[128X[104X
  [4X[32X[104X
  
  [1X49.4-4 KuKGenerators[101X
  
  [33X[1;0Y[29X[2XKuKGenerators[102X( [3XG[103X, [3Xbeta[103X, [3Xalpha[103X ) [32X function[133X
  
  [33X[0;0YIf  [3Xbeta[103X is a homomorphism from [3XG[103X into a transitive permutation group, [22XU[122X the
  full preimage of the point stabilizer and [3Xalpha[103X a homomorphism defined on (a
  superset)  of  [22XU[122X,  this  function returns images of the generators of [3XG[103X when
  mapping  to  the  wreath  product  [22X(U  [3Xalpha[103X)  ≀  ([3XG[103X  [3Xbeta[103X)[122X.  (This  is  the
  Krasner-Kaloujnine embedding theorem.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3,4),(1,2));;[127X[104X
    [4X[25Xgap>[125X [27Xhom:=GroupHomomorphismByImages(g,Group((1,2)),[127X[104X
    [4X[25X>[125X [27XGeneratorsOfGroup(g),[(1,2),(1,2)]);;[127X[104X
    [4X[25Xgap>[125X [27Xu:=PreImage(hom,Stabilizer(Image(hom),1));[127X[104X
    [4X[28XGroup([ (2,3,4), (1,2,4) ])[128X[104X
    [4X[25Xgap>[125X [27Xhom2:=GroupHomomorphismByImages(u,Group((1,2,3)),[127X[104X
    [4X[25X>[125X [27XGeneratorsOfGroup(u),[ (1,2,3), (1,2,3) ]);;[127X[104X
    [4X[25Xgap>[125X [27XKuKGenerators(g,hom,hom2);[127X[104X
    [4X[28X[ (1,4)(2,5)(3,6), (1,6)(2,4)(3,5) ][128X[104X
  [4X[32X[104X
  
  [1X49.4-5 ListWreathProductElement[101X
  
  [33X[1;0Y[29X[2XListWreathProductElement[102X( [3XG[103X, [3Xx[103X[, [3XtestDecomposition[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XListWreathProductElementNC[102X( [3XG[103X, [3Xx[103X, [3XtestDecomposition[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3Xx[103X  be  an element of a wreath product [3XG[103X where [22XG = K ≀ H[122X and [22XH[122X acts as a
  finite  permutation  group of degree [22Xm[122X. We can identify the element [3Xx[103X with a
  tuple  [22X(f_1, ..., f_m; h)[122X, where [22Xf_i ∈ K[122X is the [22Xi[122X-th base component of [3Xx[103X and
  [22Xh ∈ H[122X is the top component of [3Xx[103X.[133X
  
  [33X[0;0Y[2XListWreathProductElement[102X  returns  a  list [22X[f_1, ..., f_m, h][122X containing the
  components of [3Xx[103X or [9Xfail[109X if [3Xx[103X cannot be decomposed in the wreath product.[133X
  
  [33X[0;0YIf   omitted,   the   argument   [3XtestDecomposition[103X   defaults  to  true.  If
  [3XtestDecomposition[103X  is  true, [2XListWreathProductElement[102X makes additional tests
  to  ensure  that  the computed decomposition of [3Xx[103X is correct, i.e. it checks
  that [3Xx[103X is an element of the parent wreath product of [3XG[103X:[133X
  
  [33X[0;0YIf [22XK ≤ Sym(l)[122X, this ensures that [22Xx ∈ Sym(l) ≀ Sym(m)[122X where the parent wreath
  product  is  considered  in the same action as [3XG[103X, i.e. either in imprimitive
  action or product action.[133X
  
  [33X[0;0YIf [22XK ≤ GL(n,q)[122X, this ensures that [22Xx ∈ GL(n,q) ≀ Sym(m)[122X.[133X
  
  [1X49.4-6 WreathProductElementList[101X
  
  [33X[1;0Y[29X[2XWreathProductElementList[102X( [3XG[103X, [3Xlist[103X ) [32X function[133X
  [33X[1;0Y[29X[2XWreathProductElementListNC[102X( [3XG[103X, [3Xlist[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3Xlist[103X be equal to [22X[f_1, ..., f_m, h][122X and [3XG[103X be a wreath product where [22XG =
  K ≀ H[122X, [22XH[122X acts as a finite permutation group of degree [22Xm[122X, [22Xf_i ∈ K[122X and [22Xh ∈ H[122X.[133X
  
  [33X[0;0Y[2XWreathProductElementList[102X  returns  the element [22Xx ∈ G[122X identified by the tuple
  [22X(f_1, ..., f_m; h)[122X.[133X
  
  
  [1X49.5 [33X[0;0YFree Products[133X[101X
  
  [33X[0;0YLet  [22XG[122X  and  [22XH[122X  be  groups  with  presentations  [22X⟨  X  ∣  R ⟩[122X and [22X⟨ Y ∣ S ⟩[122X,
  respectively. Then the free product [22XG*H[122X is the group with presentation [22X⟨ X ∪
  Y  ∣ R ∪ S ⟩[122X. This construction can be generalized to an arbitrary number of
  groups.[133X
  
  
  [1X49.5-1 [33X[0;0YFreeProduct[133X[101X
  
  [33X[1;0Y[29X[2XFreeProduct[102X( [3XG[103X[, [3XH[103X, [3X...[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XFreeProduct[102X( [3Xlist[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a  finitely  presented  group  which  is the free product of the
  groups given as arguments. If the group arguments are not finitely presented
  groups, then [2XIsomorphismFpGroup[102X ([14X47.11-1[114X) must be defined for them.[133X
  
  [33X[0;0YThe operation [2XEmbedding[102X ([14X32.2-11[114X) operates on this product.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg := DihedralGroup(8);;[127X[104X
    [4X[25Xgap>[125X [27Xh := CyclicGroup(5);;[127X[104X
    [4X[25Xgap>[125X [27Xfp := FreeProduct(g,h,h);[127X[104X
    [4X[28X<fp group on the generators [ f1, f2, f3, f4, f5 ]>[128X[104X
    [4X[25Xgap>[125X [27Xfp := FreeProduct([g,h,h]);[127X[104X
    [4X[28X<fp group on the generators [ f1, f2, f3, f4, f5 ]>[128X[104X
    [4X[25Xgap>[125X [27XEmbedding(fp,2);[127X[104X
    [4X[28X[ f1 ] -> [ f4 ][128X[104X
  [4X[32X[104X
  
  
  [1X49.6 [33X[0;0YEmbeddings and Projections for Group Products[133X[101X
  
  [33X[0;0YThe  relation  between  a  group  product  and  its  factors is provided via
  homomorphisms,  the  embeddings  in the product and the projections from the
  product. Depending on the kind of product only some of these are defined.[133X
  
  [1X49.6-1 Embedding[101X
  
  [33X[1;0Y[29X[2XEmbedding[102X( [3XP[103X, [3Xnr[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  the  [3Xnr[103X-th  embedding in the group product [3XP[103X. The actual meaning of
  this  embedding  is  described  in  the  manual  section for the appropriate
  product.[133X
  
  [1X49.6-2 Projection[101X
  
  [33X[1;0Y[29X[2XProjection[102X( [3XP[103X, [3Xnr[103X ) [32X operation[133X
  
  [33X[0;0Yreturns the ([3Xnr[103X-th) projection of the group product [3XP[103X. The actual meaning of
  the  projection  returned  is  described  in  the  manual  section  for  the
  appropriate product.[133X
  
