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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,12,6}*864c
if this polytope has a name.
Group : SmallGroup(864,4406)
Rank : 5
Schlafli Type : {3,2,12,6}
Number of vertices, edges, etc : 3, 3, 12, 36, 6
Order of s₀s₁s₂s₃s₄ : 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,12,6,2} of size 1728
Vertex Figure Of :
   {2,3,2,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,6,6}*432b
   3-fold quotients : {3,2,4,6}*288a
   4-fold quotients : {3,2,6,3}*216
   6-fold quotients : {3,2,2,6}*144
   9-fold quotients : {3,2,4,2}*96
   12-fold quotients : {3,2,2,3}*72
   18-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,12,12}*1728b, {3,2,24,6}*1728c, {6,2,12,6}*1728c
Permutation Representation (GAP) :

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

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