Polytope of Type {3,12,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12,6,2}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {3,12,6,2}
Number of vertices, edges, etc : 6, 36, 72, 6, 2
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {3,4,6,2}*576, {3,12,2,2}*576
   4-fold quotients : {3,6,6,2}*432b
   9-fold quotients : {3,4,2,2}*192
   12-fold quotients : {3,2,6,2}*144, {3,6,2,2}*144
   18-fold quotients : {3,4,2,2}*96
   24-fold quotients : {3,2,3,2}*72
   36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(14,15)(17,21)(18,23)(19,22)(20,24)
(26,27)(29,33)(30,35)(31,34)(32,36);;
s1 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12)(13,17)(14,18)(15,20)(16,19)(23,24)
(25,29)(26,30)(27,32)(28,31)(35,36);;
s2 := ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,28)(14,27)(15,26)(16,25)
(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29);;
s3 := ( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)
(11,23)(12,24);;
s4 := (37,38);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(38)!( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(14,15)(17,21)(18,23)(19,22)
(20,24)(26,27)(29,33)(30,35)(31,34)(32,36);
s1 := Sym(38)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12)(13,17)(14,18)(15,20)(16,19)
(23,24)(25,29)(26,30)(27,32)(28,31)(35,36);
s2 := Sym(38)!( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,28)(14,27)(15,26)
(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29);
s3 := Sym(38)!( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)
(10,22)(11,23)(12,24);
s4 := Sym(38)!(37,38);
poly := sub<Sym(38)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 >;

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