Polytope of Type {3,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46101)
Rank : 4
Schlafli Type : {3,6,4}
Number of vertices, edges, etc : 9, 108, 144, 16
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6,4}*576b
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s2*s3*s2*s1*s3*s2*s1*s0> of order 2.
      8 facets:
         8 of {3,6}*108
      9 vertex figures:
         6 of 2-fold non-regular quotient of {6,4}*192a
         3 of 2-fold non-regular quotient of {6,4}*192a
   P/N, where N=<s2*s3*s2*s3, s1*s2*s1*s2*s3*s2*s1*s3*s2*s1> of order 4.
      4 facets:
         4 of {3,6}*108
      9 vertex figures:
         6 of 4-fold non-regular quotient of {6,4}*192a
         3 of {6,4}*48c
   P/N, where N=<s1*s2*s3*s2*s1*s3, s2*s1*s2*s3*s2*s1*s3*s2*s3> of order 4.
      4 facets:
         4 of {3,6}*108
      9 vertex figures:
         9 of 4-fold non-regular quotient of {6,4}*192a

Permutation Representation (GAP) :

s0 := ( 3, 4)( 7, 8)(11,12)(13,25)(14,26)(15,28)(16,27)(17,29)(18,30)(19,32)(20,31)(21,33)(22,34)(23,36)(24,35);;
s1 := ( 1,13)( 2,14)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,21)(10,22)(11,24)(12,23)(27,28)(31,32)(35,36);;
s2 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,21)(14,23)(15,22)(16,24)(18,19)(25,33)(26,36)(27,35)(28,34)(30,32);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36);;
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*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s0*s1*s2*s3*s2*s1*s0*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(36)!( 3, 4)( 7, 8)(11,12)(13,25)(14,26)(15,28)(16,27)(17,29)(18,30)(19,32)(20,31)(21,33)(22,34)(23,36)(24,35);
s1 := Sym(36)!( 1,13)( 2,14)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,21)(10,22)(11,24)(12,23)(27,28)(31,32)(35,36);
s2 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,21)(14,23)(15,22)(16,24)(18,19)(25,33)(26,36)(27,35)(28,34)(30,32);
s3 := Sym(36)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36);
poly := sub<Sym(36)|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*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s0*s1*s2*s3*s2*s1*s0*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
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