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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,2,6,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47319)
Rank : 6
Schlafli Type : {3,2,2,6,12}
Number of vertices, edges, etc : 3, 3, 2, 6, 36, 12
Order of s₀s₁s₂s₃s₄s₅ : 12
Order of s₀s₁s₂s₃s₄s₅s₄s₃s₂s₁ : 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) :
   2-fold quotients : {3,2,2,6,6}*864b
   3-fold quotients : {3,2,2,2,12}*576
   4-fold quotients : {3,2,2,6,3}*432
   6-fold quotients : {3,2,2,2,6}*288
   9-fold quotients : {3,2,2,2,4}*192
   12-fold quotients : {3,2,2,2,3}*144
   18-fold quotients : {3,2,2,2,2}*96
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := (4,5);;
s3 := ( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)(31,32)(34,35)
(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)(61,62)(64,65)(67,68)
(70,71)(73,74)(76,77);;
s4 := ( 6,43)( 7,42)( 8,44)( 9,49)(10,48)(11,50)(12,46)(13,45)(14,47)(15,52)
(16,51)(17,53)(18,58)(19,57)(20,59)(21,55)(22,54)(23,56)(24,70)(25,69)(26,71)
(27,76)(28,75)(29,77)(30,73)(31,72)(32,74)(33,61)(34,60)(35,62)(36,67)(37,66)
(38,68)(39,64)(40,63)(41,65);;
s5 := ( 6,63)( 7,65)( 8,64)( 9,60)(10,62)(11,61)(12,66)(13,68)(14,67)(15,72)
(16,74)(17,73)(18,69)(19,71)(20,70)(21,75)(22,77)(23,76)(24,45)(25,47)(26,46)
(27,42)(28,44)(29,43)(30,48)(31,50)(32,49)(33,54)(34,56)(35,55)(36,51)(37,53)
(38,52)(39,57)(40,59)(41,58);;
poly := Group([s0,s1,s2,s3,s4,s5]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s0*s1*s0*s1*s0*s1, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(77)!(2,3);
s1 := Sym(77)!(1,2);
s2 := Sym(77)!(4,5);
s3 := Sym(77)!( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)(31,32)
(34,35)(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)(61,62)(64,65)
(67,68)(70,71)(73,74)(76,77);
s4 := Sym(77)!( 6,43)( 7,42)( 8,44)( 9,49)(10,48)(11,50)(12,46)(13,45)(14,47)
(15,52)(16,51)(17,53)(18,58)(19,57)(20,59)(21,55)(22,54)(23,56)(24,70)(25,69)
(26,71)(27,76)(28,75)(29,77)(30,73)(31,72)(32,74)(33,61)(34,60)(35,62)(36,67)
(37,66)(38,68)(39,64)(40,63)(41,65);
s5 := Sym(77)!( 6,63)( 7,65)( 8,64)( 9,60)(10,62)(11,61)(12,66)(13,68)(14,67)
(15,72)(16,74)(17,73)(18,69)(19,71)(20,70)(21,75)(22,77)(23,76)(24,45)(25,47)
(26,46)(27,42)(28,44)(29,43)(30,48)(31,50)(32,49)(33,54)(34,56)(35,55)(36,51)
(37,53)(38,52)(39,57)(40,59)(41,58);
poly := sub<Sym(77)|s0,s1,s2,s3,s4,s5>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5, 
s3*s5*s3*s5, s0*s1*s0*s1*s0*s1, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;

to this polytope