  
  [1X20 [33X[0;0YCat-1-groups[133X[101X
  
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XAutomorphismGroupAsCatOneGroup(G)[110X[133X
  
  [33X[0;0YInputs  a group [22XG[122X and returns the Cat-1-group [22XC[122X corresponding to the crossed
  module [22XG→ Aut(G)[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XHomotopyGroup(C,n)[110X[133X
  
  [33X[0;0YInputs  a  cat-1-group [22XC[122X and an integer n. It returns the [22Xn[122Xth homotopy group
  of [22XC[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XHomotopyModule(C,2)[110X[133X
  
  [33X[0;0YInputs  a  cat-1-group  [22XC[122X and an integer n=2. It returns the second homotopy
  group  of  [22XC[122X  as  a  G-module  (i.e.  abelian  G-outer group) where G is the
  fundamental group of C.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XQuasiIsomorph(C)[110X[133X
  
  [33X[0;0YInputs  a  cat-1-group  [22XC[122X and returns a cat-1-group [22XD[122X for which there exists
  some homomorphism [22XC→ D[122X that induces isomorphisms on homotopy groups.[133X
  
  [33X[0;0YThis function was implemented by [12XLe Van Luyen[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XModuleAsCatOneGroup(G,alpha,M)[110X[133X
  
  [33X[0;0YInputs  a  group  [22XG[122X,  an abelian group [22XM[122X and a homomorphism [22Xα: G→ Aut(M)[122X. It
  returns the Cat-1-group [22XC[122X corresponding th the zero crossed module [22X0: M→ G[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XMooreComplex(C)[110X[133X
  
  [33X[0;0YInputs a cat-1-group [22XC[122X and returns its Moore complex as a G-complex (i.e. as
  a complex of groups considered as 1-outer groups).[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XNormalSubgroupAsCatOneGroup(G,N)[110X[133X
  
  [33X[0;0YInputs  a  group  [22XG[122X  with  normal  subgroup  [22XN[122X. It returns the Cat-1-group [22XC[122X
  corresponding th the inclusion crossed module [22XN→ G[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XXmodToHAP(C)[110X[133X
  
  [33X[0;0YInputs  a  cat-1-group  [22XC[122X  obtained  from  the  Xmod  package  and returns a
  cat-1-group [22XD[122X for which IsHapCatOneGroup(D) returns true.[133X
  
  [33X[0;0YIt returns "fail" id [22XC[122X has not been produced by the Xmod package.[133X
  
