Polytope of Type {4,12,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,3}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47847)
Rank : 4
Schlafli Type : {4,12,3}
Number of vertices, edges, etc : 12, 144, 108, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   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) :
   4-fold quotients : {4,6,3}*432b
   9-fold quotients : {4,4,3}*192b
   18-fold quotients : {2,4,3}*96
   36-fold quotients : {4,2,3}*48, {2,4,3}*48
   72-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      4 facets:
         2 of {4,6}*144
         2 of {4,12}*288
      12 vertex figures:
         12 of 2-fold non-regular quotient of {12,3}*144
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      6 facets:
         6 of 3-fold non-regular quotient of {4,12}*288
      4 vertex figures:
         4 of {12,3}*144
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 3.
      6 facets:
         6 of 3-fold non-regular quotient of {4,12}*288
      8 vertex figures:
         2 of {12,3}*144
         6 of {4,3}*48
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 6.
      4 facets:
         2 of 3-fold non-regular quotient of {4,6}*144
         2 of 3-fold non-regular quotient of {4,12}*288
      8 vertex figures:
         2 of 2-fold non-regular quotient of {12,3}*144
         6 of 2-fold non-regular quotient of {4,3}*48
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 6.
      4 facets:
         2 of 3-fold non-regular quotient of {4,6}*144
         2 of 3-fold non-regular quotient of {4,12}*288
      4 vertex figures:
         4 of 2-fold non-regular quotient of {12,3}*144

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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