Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*432b
Also Known As : {{6,6|2},{6,6|2}}. if this polytope has another name.
Group : SmallGroup(432,759)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 18, 18, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,6,2} of size 864
   {6,6,6,3} of size 1296
   {6,6,6,4} of size 1728
   {6,6,6,3} of size 1728
   {6,6,6,4} of size 1728
Vertex Figure Of :
   {2,6,6,6} of size 864
   {3,6,6,6} of size 1296
   {4,6,6,6} of size 1728
   {3,6,6,6} of size 1728
   {4,6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a
   6-fold quotients : {3,2,6}*72, {6,2,3}*72
   9-fold quotients : {2,2,6}*48, {2,6,2}*48, {6,2,2}*48
   12-fold quotients : {3,2,3}*36
   18-fold quotients : {2,2,3}*24, {2,3,2}*24, {3,2,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6,12}*864b, {6,12,6}*864b, {12,6,6}*864b
   3-fold covers : {6,6,18}*1296b, {6,18,6}*1296a, {18,6,6}*1296b, {6,6,6}*1296g, {6,6,6}*1296j, {6,6,6}*1296q, {6,6,6}*1296s
   4-fold covers : {6,6,24}*1728b, {6,24,6}*1728b, {24,6,6}*1728b, {12,6,12}*1728b, {6,12,12}*1728b, {12,12,6}*1728b, {6,6,12}*1728a, {6,12,6}*1728e, {6,12,6}*1728i, {12,6,6}*1728a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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