Polytope of Type {3,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,14}*1512
Also Known As : {{3,6}₆,{6,14|2}}. if this polytope has another name.
Group : SmallGroup(1512,486)
Rank : 4
Schlafli Type : {3,6,14}
Number of vertices, edges, etc : 9, 27, 126, 14
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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 : {3,6,14}*504
   7-fold quotients : {3,6,2}*216
   9-fold quotients : {3,2,14}*168
   18-fold quotients : {3,2,7}*84
   21-fold quotients : {3,6,2}*72
   63-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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