Polytope of Type {6,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*288a
Also Known As : {{6,6|2},{6,4|2}}. if this polytope has another name.
Group : SmallGroup(288,958)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 6, 18, 12, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,4,2} of size 576
   {6,6,4,4} of size 1152
   {6,6,4,6} of size 1728
   {6,6,4,3} of size 1728
Vertex Figure Of :
   {2,6,6,4} of size 576
   {3,6,6,4} of size 864
   {4,6,6,4} of size 1152
   {3,6,6,4} of size 1152
   {4,6,6,4} of size 1152
   {6,6,6,4} of size 1728
   {6,6,6,4} of size 1728
   {6,6,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,2}*144a
   3-fold quotients : {2,6,4}*96a, {6,2,4}*96
   6-fold quotients : {3,2,4}*48, {2,6,2}*48, {6,2,2}*48
   9-fold quotients : {2,2,4}*32
   12-fold quotients : {2,3,2}*24, {3,2,2}*24
   18-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12,4}*576a, {12,6,4}*576a, {6,6,8}*576a
   3-fold covers : {6,18,4}*864a, {18,6,4}*864a, {6,6,4}*864b, {6,6,12}*864b, {6,6,4}*864h, {6,6,12}*864f
   4-fold covers : {12,12,4}*1152b, {6,12,8}*1152b, {6,24,4}*1152c, {6,12,8}*1152e, {6,24,4}*1152f, {6,12,4}*1152b, {12,6,8}*1152b, {24,6,4}*1152b, {6,6,16}*1152b, {6,6,4}*1152c, {6,12,4}*1152i, {12,6,4}*1152b
   5-fold covers : {6,6,20}*1440a, {6,30,4}*1440b, {30,6,4}*1440b
   6-fold covers : {36,6,4}*1728a, {12,18,4}*1728a, {18,12,4}*1728a, {6,36,4}*1728a, {12,6,4}*1728a, {6,12,4}*1728b, {6,18,8}*1728a, {18,6,8}*1728a, {6,6,8}*1728b, {6,6,24}*1728b, {12,6,12}*1728b, {12,6,12}*1728d, {6,12,12}*1728b, {6,12,12}*1728e, {6,6,8}*1728e, {6,6,24}*1728f, {6,12,4}*1728j, {12,6,4}*1728h
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)
(21,23)(26,27)(28,31)(29,33)(30,32)(35,36);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,29)(20,28)(21,30)(22,32)
(23,31)(24,33)(25,35)(26,34)(27,36);;
s3 := ( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)
(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36);;
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(36)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36);
s1 := Sym(36)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)
(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36);
s2 := Sym(36)!( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,29)(20,28)(21,30)
(22,32)(23,31)(24,33)(25,35)(26,34)(27,36);
s3 := Sym(36)!( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)
(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36);
poly := sub<Sym(36)|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, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope