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Polytope of Type {6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*432a
if this polytope has a name.
Group : SmallGroup(432,530)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 54, 108, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {6,4,2} of size 864
   {6,4,4} of size 1728
Vertex Figure Of :
   {2,6,4} of size 864
   {4,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*216
   3-fold quotients : {6,4}*144
   6-fold quotients : {6,4}*72
   27-fold quotients : {2,4}*16
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,8}*864a, {12,4}*864b
   3-fold covers : {6,4}*1296a, {6,12}*1296m, {6,12}*1296o
   4-fold covers : {6,16}*1728a, {12,4}*1728b, {12,8}*1728a, {24,4}*1728a, {24,4}*1728c, {12,8}*1728d
Permutation Representation (GAP) :
s0 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)
(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)
(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)(20,21)
(22,23)(25,27)(28,37)(29,39)(30,38)(31,41)(32,40)(33,42)(34,45)(35,44)(36,43)
(47,48)(49,50)(52,54);;
s2 := ( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,15)(16,22)(17,24)
(18,23)(25,27)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,42)(43,49)
(44,51)(45,50)(52,54);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(54)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)
(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)
(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);
s1 := Sym(54)!( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)
(20,21)(22,23)(25,27)(28,37)(29,39)(30,38)(31,41)(32,40)(33,42)(34,45)(35,44)
(36,43)(47,48)(49,50)(52,54);
s2 := Sym(54)!( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,15)(16,22)
(17,24)(18,23)(25,27)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,42)
(43,49)(44,51)(45,50)(52,54);
poly := sub<Sym(54)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 >; 
 
References : None.
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