  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YLet   [23X\Omega[123X  be  a  set  of  cardinality  [23Xd\in\mathbb{N}_{\ge  3}[123X  and  let
  [23XT_{d}=(V,E)[123X  be  the [23Xd[123X-regular tree. We follow Serre's graph theory notation
  [Ser80]. Given a subgroup [23XH[123X of the automorphism group [23X\mathrm{Aut}(T_{d})[123X of
  [23XT_{d}[123X,  and  a  vertex  [23Xx\in  V[123X,  the  stabilizer  [23XH_{x}[123X of [23Xx[123X in [23XH[123X induces a
  permutation  group  on  the set [23XE(x):=\{e\in E\mid o(e)=x\}[123X of edges issuing
  from  [23Xx[123X.  We  say that [23XH[123X is locally "P" if for every [23Xx\in V[123X said permutation
  group  satisfies  the  property  "P",  e.g. being transitive, semiprimitive,
  quasiprimitive or [23X2[123X-transitive.[133X
  
  [33X[0;0YIn  [BM00],  Burger-Mozes  develop  a remarkable structure theory of closed,
  non-discrete, locally quasiprimitive subgroups of [23X\mathrm{Aut}(T_{d})[123X, which
  resembles  the  theory  of  semisimple  Lie  groups.  They  complement  this
  structure  theory  with  a  particularly  accessible  class  of subgroups of
  [23X\mathrm{Aut}(T_{d})[123X      with     prescribed     local     action:     Given
  [23XF\le\mathrm{Sym}(\Omega)[123X,           their           universal          group
  [23X\mathrm{U}(F)\le\mathrm{Aut}(T_{d})[123X    is   closed,   compactly   generated,
  vertex-transitive and locally permutation isomorphic to [23XF[123X. It is discrete if
  and only if [23XF[123X is semiregular. When [23XF[123X is transitive, [23X\mathrm{U}(F)[123X is maximal
  up  to  conjugation among vertex-transitive subgroups of [23X\mathrm{Aut}(T_{d})[123X
  that are locally permutation isomorphic to [23XF[123X, hence [13Xuniversal[113X.[133X
  
  [33X[0;0YThis  construction  was  generalized by the second author in [Tor20]: In the
  spirit  of  [23Xk[123X-closures  of  groups  acting on trees developed in [BEW15], we
  generalize  the universal group construction by prescribing the local action
  on  balls  of  a  given radius [23Xk\in\mathbb{N}[123X, the Burger-Mozes construction
  corresponding  to  the case [23Xk=1[123X. Fix a tree [23XB_{d,k}[123X which is isomorphic to a
  ball  of  radius  [23Xk[123X  in  the labelled tree [23XT_{d}[123X and let [23Xl_{x}^{k}:B(x,k)\to
  B_{d,k}[123X ([23Xx\in V[123X) be the unique label-respecting isomorphism. Then[133X
  
  
  [24X[33X[0;6Y\sigma_{k}:\mathrm{Aut}(T_{d})\times   V\to\mathrm{Aut}(B_{d,k}),\  (g,x)\to
  l_{gx}^{k}\circ g\circ (l_{x}^{k})^{-1}[133X
  
  [124X
  
  [33X[0;0Ycaptures the [13X[23Xk[123X-local action[113X of [23Xg[123X at the vertex [23Xx\in V[123X.[133X
  
  [33X[0;0YWith     this    we    can    make    the    following    definition:    Let
  [23XF\!\le\!\mathrm{Aut}(B_{d,k})[123X. Define[133X
  
  
  [24X[33X[0;6Y\mathrm{U}_{k}(F):=\{g\in\mathrm{Aut}(T_{d})\mid     \forall     x\in    V:\
  \sigma_{k}(g,x)\in F\}.[133X
  
  [124X
  
  [33X[0;0YWhile  [23X\mathrm{U}_{k}(F)[123X  is  always closed, vertex-transitive and compactly
  generated,  other  properties  of [23X\mathrm{U}(F)[123X do [13Xnot[113X carry over. Foremost,
  the  group  [23X\mathrm{U}_{k}(F)[123X need not be locally action isomorphic to [23XF[123X and
  we say that [23XF\le\mathrm{Aut}(B_{d,k})[123X satisfies condition (C) if it is. This
  can  be  viewed  as  an  interchangeability  condition on neighbouring local
  actions,  see  Section  [14X3.1[114X.  There  is also a discreteness condition (D) on
  [23XF\le\mathrm{Aut}(B_{d,k})[123X  in  terms of certain stabilizers in [23XF[123X under which
  [23X\mathrm{U}_{k}(F)[123X is discrete, see Section [14X5.1[114X.[133X
  
  [33X[0;0Y[5XUGALY[105X   provides   methods   to  create,  analyse  and  find  local  actions
  [23XF\le\mathrm{Aut}(B_{d,k})[123X  that  satisfy condition (C) and/or (D), including
  the  constructions  [23X\Gamma[123X,  [23X\Delta[123X,  [23X\Phi[123X,  [23X\Sigma[123X,  and  [23X\Pi[123X  developed in
  [Tor20].  This  package  was  developed within the Zero-Dimensional Symmetry
  Research   Group   ([7Xhttps://zerodimensional.group/"[107X)   in   the   School  of
  Mathematical               and               Physical               Sciences
  ([7Xhttps://www.newcastle.edu.au/school/mathematical-and-physical-sciences[107X)  at
  The  University  of  Newcastle  ([7Xhttps://www.newcastle.edu.au/[107X) as part of a
  project course taken by the first author, supervised by the second author.[133X
  
  
  [1X1.1 [33X[0;0YPurpose[133X[101X
  
  [33X[0;0YNote:  many of the examples in this manual access random elements of various
  domains   via   [10XRandom()[110X.  To  ensure  reproducibility  and  testability  we
  initialize the random source [10Xmt[110X below each time.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmt:=RandomSource(IsMersenneTwister,1);[127X[104X
    [4X[28X<RandomSource in IsMersenneTwister>[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XUGALY[105X  serves  both a research and an educational purpose. It consolidates a
  rudimentary  codebase  that was developed by the second author in the course
  of  research  undertaken towards the article [Tor20]. This codebase had been
  tremendously  beneficial  in  achieving  the results of [Tor20] in the first
  place  and so there has always been a desire to make it available to a wider
  audience.[133X
  
  [33X[0;0YFrom  a  research perspective, [5XUGALY[105X introduces computational methods to the
  world  of locally compact groups. Due to the Cayley-Abels graph construction
  [KM08],  groups  acting  on  trees  form a particularly significant class of
  totally  disconnected  locally compact groups. Burger-Mozes universal groups
  [BM00]     and     their     generalisations     [23X\mathrm{U}_{k}(F)[123X,    where
  [23XF\le\mathrm{Aut}(B_{d,k})[123X  satisfies  the  compatibility  condition (C), are
  among  the  most accessible of these groups and form a significant subclass:
  in fact, due to [Tor20, Corollary 4.32], the locally transitive, generalised
  universal  groups  are  exactly  the closed, locally transitive subgroups of
  [23X\mathrm{Aut}(T_{d})[123X  that contain an inversion of order [23X2[123X and satisfy one of
  the  independence  properties  [23X(P_{k})[123X  (see  [BEW15]) that generalise Tits'
  independence  property  [23X(P)[123X, see [Tit70]. Subgroups of [23X\mathrm{Aut}(B_{d,k})[123X
  are  treated  as objects of the category [2XIsLocalAction[102X ([14X2.1-1[114X) to the effect
  that  they  remember the degree [23Xd[123X the radius [23Xk[123X of the tree [23XB_{d,k}[123X that they
  act  on as a permutation group on its [23Xd\cdot(d-1)^{k-1}[123X leaves. For example,
  the automorphism group of [23XB_{3,2}[123X can be accessed as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=AutBall(3,2);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(F);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalActionDegree(F);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XLocalActionRadius(F);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  general, a subgroup [23XF[123X of the permutation group [23X\mathrm{Aut}(B_{d,k})[123X can
  be  turned  into  an object of the category [2XIsLocalAction[102X ([14X2.1-1[114X) by calling
  the  creator  operation  [2XLocalAction[102X ([14X2.1-2[114X) with the degree [23Xd[123X, the radius [23Xk[123X
  and   the   permutation   group   [23XF[123X   itself.   For  example,  the  subgroup
  [23XA_{3}\le\mathrm{Aut}(B_{3,1})\cong S_{3}[123X can be generated as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA3:=LocalAction(3,1,AlternatingGroup(3));[127X[104X
    [4X[28XAlt( [ 1 .. 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(A3);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalActionDegree(A3);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XLocalActionRadius(A3);[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XUGALY[105X  provides the means to generate a library of all generalised universal
  groups    [23X\mathrm{U}_{k}(F)[123X    in    terms    of    their   [23Xk[123X-local   action
  [23XF\le\mathrm{Aut}(B_{d,k})[123X  satisfying  the  compatibility  condition (C). We
  envision  to  add such a library in a future version of this package. In the
  case  [23Xk=1[123X  of classical Burger-Mozes groups, the compatibility condition (C)
  is  void  and  so  the  library  would  coincide  with the library of finite
  transitive permutation groups [5XTransGrp[105X. For example, in the case [23X(d,k)=(3,1)[123X
  there  are  only  two  local  actions,  corresponding  to the two transitive
  permutation groups of degree [23X3[123X, namely [23XA_{3}[123X and [23XS_{3}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA3:=LocalAction(3,1,TransitiveGroup(3,1));[127X[104X
    [4X[28XA3[128X[104X
    [4X[25Xgap>[125X [27XS3:=LocalAction(3,1,TransitiveGroup(3,2));[127X[104X
    [4X[28XS3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo  create  this  library for the case [23X(d,k)=(3,2)[123X we organise the subgroups
  [23XF\le\mathrm{Aut}(B_{3,2})[123X  that  satisfy  the  compatibility  condition  (C)
  according  to  which subgroup of [23X\mathrm{Aut}(B_{3,1})[123X they project to under
  the  natural  projection  [23X\mathrm{Aut}(B_{3,2})\to\mathrm{Aut}(B_{3,1})[123X that
  restricts  automorphisms  to  [23XB_{3,1}\subseteq  B_{3,2}[123X.  In other words, we
  organise the subgroups [23XF\le\mathrm{Aut}(B_{3,2})[123X satisfying (C) according to
  [23X\sigma_{1}(F,b)\le\mathrm{Aut}(B_{3,1})[123X.                               Using
  [2XConjugacyClassRepsCompatibleGroupsWithProjection[102X ([14X3.3-5[114X), the following code
  illustrates  that  there  is  one conjugacy class of groups that projects to
  [23XA_{3}[123X whereas five project to [23XS_{3}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA3_extn:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,A3);[127X[104X
    [4X[28X[ Group([ (1,4,5)(2,3,6) ]) ][128X[104X
    [4X[25Xgap>[125X [27XS3_extn:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3);[127X[104X
    [4X[28X[ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), [128X[104X
    [4X[28X  Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), [128X[104X
    [4X[28X  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]), [128X[104X
    [4X[28X  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]), [128X[104X
    [4X[28X  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  of these groups have been identified to stem from general constructions
  of  groups [23X\widetilde{F}\le\mathrm{Aut}(B_{d,2})[123X satisfying (C) from a given
  group  [23XF\le\mathrm{Aut}(B_{d,1})[123X,  much  like  some finite transitive groups
  have   been   organised   into  families.  Specifically,  the  constructions
  [23X\Gamma(F)[123X,  [23X\Delta(F)[123X,  [23X\Pi(F,\rho,X)[123X  and [23X\Phi(F)[123X introduced in the article
  [Tor20,   Section   3.4]   can   be   accessed   via   the  [5XUGALY[105X  functions
  [2XLocalActionGamma[102X  ([14X4.1-2[114X),  [2XLocalActionDelta[102X  ([14X4.1-3[114X), [2XLocalActionPi[102X ([14X4.4-4[114X)
  and  [2XLocalActionPhi[102X ([14X4.2-1[114X) respectively, see Chapter [14X4[114X. Below, we use these
  functions to identify all six groups of the previous output.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLocalActionPhi(A3)=A3_extn[1];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalActionGamma(3,S3)=S3_extn[1];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalActionDelta(3,S3)=S3_extn[2];[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsConjugate(AutBall(3,2),LocalActionDelta(3,S3),S3_extn[2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrho:=SignHomomorphism(S3);;[127X[104X
    [4X[25Xgap>[125X [27XLocalActionPi(2,3,S3,rho,[0,1])=S3_extn[3];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalActionPi(2,3,S3,rho,[1])=S3_extn[4];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalActionPhi(S3)=S3_extn[5];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XUGALY[105X  may  also  be  a  useful  tool in the context of the Weiss conjecture
  [Wei78],  which  in  particular  states  that  there  are only finitely many
  conjugacy  classes  of  discrete,  vertex-transitive  and  locally primitive
  subgroup  of [23X\mathrm{Aut}(T_{d})[123X. When such a group contains an inversion of
  order  [23X2[123X,  it  can  be written as a universal group [23X\mathrm{U}_{k}(F)[123X, where
  [23XF\le\mathrm{Aut}(B_{d,k})[123X satisfies both the compatibility condition (C) and
  the  discreteness  condition (D), due to [Tor20, Corollary 4.38]. Therefore,
  [5XUGALY[105X  can  be used to construct explicit examples of groups relevant to the
  Weiss  conjecture.  Their  structure as well as patterns in their appearance
  may  provide  more  insight  into  the  conjecture and suggest directions of
  research.  At  the very least, [5XUGALY[105X provides lower bounds on their numbers.
  For  example,  consider the case [23Xd=4[123X. There are exactly two primitive groups
  of  degree  [23X4[123X, namely [23XA_{4}[123X and [23XS_{4}[123X, which we readily turn into objects of
  the category [2XIsLocalAction[102X ([14X2.1-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrPrimitiveGroups(4);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XA4:=LocalAction(4,1,PrimitiveGroup(4,1));;[127X[104X
    [4X[25Xgap>[125X [27XS4:=LocalAction(4,1,PrimitiveGroup(4,2));;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext,  we  proceed  as  before  to  determine  how many conjugacy classes of
  subgroups  of  [23X\mathrm{Aut}(B_{4,2})[123X  with  (C)  there are that project onto
  [23XA_{4}[123X  and [23XS_{4}[123X respectively. We then filter the output for subgroups that,
  in addition, satisfy the discreteness condition (D), see [2XSatisfiesD[102X ([14X5.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA4_extn:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,A4);;[127X[104X
    [4X[25Xgap>[125X [27XSize(A4_extn); Size(Filtered(A4_extn,SatisfiesD));[127X[104X
    [4X[28X5[128X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XS4_extn:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,S4);;[127X[104X
    [4X[25Xgap>[125X [27XSize(S4_extn); Size(Filtered(S4_extn,SatisfiesD));[127X[104X
    [4X[28X13[128X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [23XA_{4}[123X  there  are  two, and for [23XS_{4}[123X there are three. We conclude that
  there  are  at  least [23X5=2+3[123X conjugacy classes of discrete, vertex-transitive
  and  locally  primitive subgroups of [23X\mathrm{Aut}(T_{4})[123X. More examples, and
  hence a better lower bound, can be obtained by increasing [23Xk[123X.[133X
  
  [33X[0;0YEvery  subgroup  [23XF\le\mathrm{Aut}(B_{d,k})[123X  which satisfies both (C) and (D)
  admits  an  involutive  compatibility  cocycle (see [Tor20, Section 3.2.2]),
  i.e.  a  map [23Xz:F\times\{1,\ldots,d\}\to F[123X which satisfies certain properties
  reflecting the discreteness of the group [23X\mathrm{U}_{k}(F)[123X. It is intriguing
  that some groups [23XF\le\mathrm{Aut}(B_{d,k})[123X with (C) and (D) stem from groups
  [23XF'\le\mathrm{Aut}(B_{d,k-1})[123X   that   satisfy   (C),   admit  an  involutive
  compatibility  cocycle  [23Xz[123X  but  do  not  satisfy  (D),  in  the sense of the
  construction  [23XF=\Gamma_{z}(F')[123X  (see  [Tor20,  Proposition  3.26]),  whereas
  others  do  not.  For  example, in the case [23Xd=3[123X, five of the seven conjugacy
  classes  of  discrete,  vertex-transitive and locally primitive subgroups of
  [23X\mathrm{Aut}(T_{3})[123X  come  from generalised universal groups. Of these five,
  three  arise  from  groups  [23XF'[123X  as above while the remaining two do not, see
  [Tor20,  Example 4.39]. The three groups are [23X\Gamma(A_{3})[123X and [23X\Gamma(S_{3})[123X
  and   [23X\Gamma_{z}(\Pi(S_{3},\mathrm{sgn},\{1\}))[123X.   The  code  example  below
  verifies  that  [23X\Pi(S_{3},\mathrm{sgn},\{1\})\le\mathrm{Aut}(B_{3,2})[123X indeed
  satisfies  (C),  does not satisfy (D) but admits an involutive compatibility
  cocycle  [23Xz[123X,  which  can  be  obtained  using  [2XInvolutiveCompatibilityCocycle[102X
  ([14X5.3-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS3:=SymmetricGroup(3);;[127X[104X
    [4X[25Xgap>[125X [27Xrho:=SignHomomorphism(S3);;[127X[104X
    [4X[25Xgap>[125X [27XH:=LocalActionPi(2,3,S3,rho,[1]);;[127X[104X
    [4X[25Xgap>[125X [27X[SatisfiesC(H), SatisfiesD(H), not InvolutiveCompatibilityCocycle(H)=fail];[127X[104X
    [4X[28X[ true, false, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  then  find  that  there  are  four  conjugacy  classes  of  subgroups of
  [23X\mathrm{Aut}(B_{3,3})[123X     that     satisfy     (C)    and    project    onto
  [23X\Pi(S_{3},\mathrm{sgn},\{1\})[123X    under    the    natural    projection   map
  [23X\mathrm{Aut}(B_{3,3})\to\mathrm{Aut}(B_{3,2})[123X.  Of  these  four  groups, two
  also       satisy       (D)      and      one      is      conjugate      to
  [23X\Gamma_{z}(\Pi(S_{3},\mathrm{sgn},\{1\}))[123X,    which   we   construct   using
  [2XLocalActionGamma[102X ([14X4.1-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgrps:=ConjugacyClassRepsCompatibleGroupsWithProjection(3,H);; Size(grps);[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XSize(Filtered(grps,SatisfiesD));[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xz:=InvolutiveCompatibilityCocycle(H);;[127X[104X
    [4X[25Xgap>[125X [27XSize(Intersection(LocalActionGamma(H,z)^AutBall(3,3),grps));[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  number  of  different  (involutive) compatibility cocycles that a group
  [23XF\le\mathrm{Aut}(B_{d,k})[123X  may  admit  is  also mysterious, including in the
  case  [23Xk=1[123X. For example, consider the case [23X(d,k)=(4,1)[123X. We compute the set of
  all  involutive  compatibility cocycles of a local action using the function
  [2XAllInvolutiveCompatibilityCocycles[102X ([14X5.3-2[114X):[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgrps:=AllTransitiveGroups(NrMovedPoints,4);[127X[104X
    [4X[28X[ C(4) = 4, E(4) = 2[x]2, D(4), A4, S4 ][128X[104X
    [4X[25Xgap>[125X [27XApply(grps,H->Size(AllInvolutiveCompatibilityCocycles(LocalAction(4,1,H))));;[127X[104X
    [4X[25Xgap>[125X [27Xgrps;[127X[104X
    [4X[28X[ 1, 1, 8, 28, 256 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFrom  an  educational point of view, we envision that [5XUGALY[105X could be used to
  enhance  the learning experience of students in the area of groups acting on
  trees.  The  class  of generalised universal groups forms an ideal framework
  for  this  purpose.  For  example, to internalise the widely used concept of
  local  actions  it may be helpful to take a [23X2[123X-local action in the form of an
  automorphism  of [23XB_{3,2}[123X, decompose it into its [23X1[123X-local actions, and recover
  the  original  autmorphism  from them: in the example below, we start with a
  random  automorphism  [10Xaut[110X of [23XB_{3,2}[123X. We then compute its [23X1[123X-local actions at
  the  center vertex, represented by the address [10X[][110X, as well as its neighbours
  [10X[1][110X, [10X[2][110X and [10X[3][110X using [2XLocalAction[102X ([14X2.1-6[114X). Finally, we recover [10Xaut[110X from the
  [23X1[123X-local  actions  at  the  center's  neighbours  using  [2XAssembleAutomorphism[102X
  ([14X3.2-4[114X), which only requires the local actions at the center's neighbours.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmt:=RandomSource(IsMersenneTwister,1);;[127X[104X
    [4X[25Xgap>[125X [27Xaut:=Random(mt,AutBall(3,2));[127X[104X
    [4X[28X(1,4,5,2,3,6)[128X[104X
    [4X[25Xgap>[125X [27Xaut_center:=LocalAction(1,3,2,aut,[]);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27Xaut_1:=LocalAction(1,3,2,aut,[1]);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27Xaut_2:=LocalAction(1,3,2,aut,[2]);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27Xaut_3:=LocalAction(1,3,2,aut,[3]);[127X[104X
    [4X[28X(1,3)[128X[104X
    [4X[25Xgap>[125X [27XAssembleAutomorphism(3,1,[aut_1,aut_2,aut_3]);[127X[104X
    [4X[28X(1,4,5,2,3,6)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  computationally  inclined  student  may  also  benefit  from  verifying
  existing  theorems  using  [5XUGALY[105X.  For  example, one way to phrase a part of
  Tutte's  work  [Tut47] [Tut59] is to say that there are only three conjugacy
  classes  of  discrete,  locally  transitive subgroups of [23X\mathrm{Aut}(T_{3})[123X
  that  contain  an  inversion of order [23X2[123X and are [23XP_{2}[123X-closed. Due to [Tor20,
  Corollary  4.38],  this  can  be verified by checking that among all locally
  transitive    subgroups   of   [23X\mathrm{Aut}(B_{3,2})[123X   which   satisfy   the
  compatibility  condition  (C),  only  three  also  satisfy  the discreteness
  condition  (D). In the code example below, we start this task by turning the
  two  transitive  groups of degree [23X3[123X, namely [23XA_{3}[123X and [23XS_{3}[123X, into objects of
  the  category  [2XIsLocalAction[102X ([14X2.1-1[114X). For each of them we proceed to compute
  the  list of subgroups of [23X\mathrm{Aut}(B_{3,2})[123X that satisfy (C) and project
  onto  the respective group as before. Now we merely have to go through these
  lists  and  check  whether or not condition (D) is satisfied. Indeed we find
  exactly three groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA3:=LocalAction(3,1,TransitiveGroup(3,1));;[127X[104X
    [4X[25Xgap>[125X [27XS3:=LocalAction(3,1,TransitiveGroup(3,2));;[127X[104X
    [4X[25Xgap>[125X [27XA3_extn:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,A3);[127X[104X
    [4X[28X[ Group([ (1,4,5)(2,3,6) ]) ][128X[104X
    [4X[25Xgap>[125X [27XS3_extn:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3);[127X[104X
    [4X[28X[ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), [128X[104X
    [4X[28X  Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), [128X[104X
    [4X[28X  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]), [128X[104X
    [4X[28X  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]), [128X[104X
    [4X[28X  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ][128X[104X
    [4X[25Xgap>[125X [27XApply(A3_extn,SatisfiesD); A3_extn;[127X[104X
    [4X[28X[ true ][128X[104X
    [4X[25Xgap>[125X [27XApply(S3_extn,SatisfiesD); S3_extn;[127X[104X
    [4X[28X[ true, true, false, false, false ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  may  also  be  instructive to generate involutive compatibility cocycles
  computationally  and  check  parts  of  the  axioms manually. In the example
  below,          we          first         generate         the         group
  [23X\Pi(S_{3},\mathrm{sgn},\{1\})\le\mathrm{Aut}(B_{3,2})[123X,  which we know admits
  an  involutive  compatibility  cocycle  from before. We then check that [23Xz[123X is
  indeed  involutive on a random element [10Xa[110X [23X\in\Pi(S_{3},\mathrm{sgn},\{1\})[123X in
  direction [23X1[123X by checking that [23Xz(z(a,1),1)=a[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS3:=SymmetricGroup(3);;[127X[104X
    [4X[25Xgap>[125X [27Xrho:=SignHomomorphism(S3);;[127X[104X
    [4X[25Xgap>[125X [27XH:=LocalActionPi(2,3,S3,rho,[1]);;[127X[104X
    [4X[25Xgap>[125X [27Xz:=InvolutiveCompatibilityCocycle(H);;[127X[104X
    [4X[25Xgap>[125X [27Xmt:=RandomSource(IsMersenneTwister,1);;[127X[104X
    [4X[25Xgap>[125X [27Xa:=Random(mt,H); Image(z,[Image(z,[a,1]),1]);[127X[104X
    [4X[28X(1,5,3)(2,6,4)[128X[104X
    [4X[28X(1,5,3)(2,6,4)[128X[104X
  [4X[32X[104X
  
