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Polytope of Type {4,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,6}*864b
Also Known As : {{4,4}6,{4,6|2}}. if this polytope has another name.
Group : SmallGroup(864,4686)
Rank : 4
Schlafli Type : {4,4,6}
Number of vertices, edges, etc : 18, 36, 54, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,6,2} of size 1728
Vertex Figure Of :
   {2,4,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,4,2}*288
   6-fold quotients : {4,4,2}*144
   18-fold quotients : {2,2,6}*48
   36-fold quotients : {2,2,3}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,12}*1728b, {4,4,6}*1728b
Permutation Representation (GAP) :
s0 := ( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(31,37)
(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51);;
s1 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)
(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);;
s2 := ( 1,13)( 2,15)( 3,14)( 5, 6)( 7,22)( 8,24)( 9,23)(11,12)(16,19)(17,21)
(18,20)(26,27)(28,40)(29,42)(30,41)(32,33)(34,49)(35,51)(36,50)(38,39)(43,46)
(44,48)(45,47)(53,54);;
s3 := ( 1,29)( 2,28)( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)
(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,47)(20,46)(21,48)
(22,50)(23,49)(24,51)(25,53)(26,52)(27,54);;
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*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, 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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(54)!( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)
(31,37)(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51);
s1 := Sym(54)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)
(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);
s2 := Sym(54)!( 1,13)( 2,15)( 3,14)( 5, 6)( 7,22)( 8,24)( 9,23)(11,12)(16,19)
(17,21)(18,20)(26,27)(28,40)(29,42)(30,41)(32,33)(34,49)(35,51)(36,50)(38,39)
(43,46)(44,48)(45,47)(53,54);
s3 := Sym(54)!( 1,29)( 2,28)( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)
(10,38)(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,47)(20,46)
(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54);
poly := sub<Sym(54)|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*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
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 >; 
 
References : None.
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