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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,12}*864a
if this polytope has a name.
Group : SmallGroup(864,4368)
Rank : 5
Schlafli Type : {3,2,6,12}
Number of vertices, edges, etc : 3, 3, 6, 36, 12
Order of s₀s₁s₂s₃s₄ : 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,6,12,2} of size 1728
Vertex Figure Of :
   {2,3,2,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,6,6}*432a
   3-fold quotients : {3,2,2,12}*288, {3,2,6,4}*288a
   6-fold quotients : {3,2,2,6}*144, {3,2,6,2}*144
   9-fold quotients : {3,2,2,4}*96
   12-fold quotients : {3,2,2,3}*72, {3,2,3,2}*72
   18-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,6,24}*1728a, {3,2,12,12}*1728a, {6,2,6,12}*1728a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope