Polytope of Type {6,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,6}*864a
Also Known As : {{6,4|2},{4,6}₄}. if this polytope has another name.
Group : SmallGroup(864,4686)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 6, 36, 36, 18
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,4,6,2} of size 1728
Vertex Figure Of :
   {2,6,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4,6}*432b
   3-fold quotients : {2,4,6}*288
   6-fold quotients : {2,4,6}*144
   9-fold quotients : {6,4,2}*96a
   18-fold quotients : {6,2,2}*48
   27-fold quotients : {2,4,2}*32
   36-fold quotients : {3,2,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,8,6}*1728b, {12,4,6}*1728a, {6,4,12}*1728b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s3*s2*s1*s2*s3*s2*s1*s2*s3> of order 2.
      9 facets:
         9 of {6,4}*48a
      6 vertex figures:
         6 of 2-fold non-regular quotient of {4,6}*144
   P/N, where N=<s2*s1*s2*s3*s2*s1*s2*s3> of order 3.
      6 facets:
         6 of {6,4}*48a
      6 vertex figures:
         6 of 3-fold non-regular quotient of {4,6}*144
   P/N, where N=<s1*s2*s3*s2*s1*s3> of order 3.
      6 facets:
         6 of {6,4}*48a
      6 vertex figures:
         6 of 3-fold non-regular quotient of {4,6}*144
   P/N, where N=<s1*s2*s3*s2*s1*s3, s2*s3*s2*s3*s2*s3> of order 6.
      3 facets:
         3 of {6,4}*48a
      6 vertex figures:
         6 of 6-fold non-regular quotient of {4,6}*144

Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54);;
s1 := ( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);;
s2 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);;
s3 := ( 1,40)( 2,41)( 3,42)( 4,37)( 5,38)( 6,39)( 7,43)( 8,44)( 9,45)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(16,34)(17,35)(18,36)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(54)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54);
s1 := Sym(54)!( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);
s2 := Sym(54)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);
s3 := Sym(54)!( 1,40)( 2,41)( 3,42)( 4,37)( 5,38)( 6,39)( 7,43)( 8,44)( 9,45)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(16,34)(17,35)(18,36)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);
poly := sub<Sym(54)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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