Polytope of Type {6,6,2,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,2,2}*864d
if this polytope has a name.
Group : SmallGroup(864,4704)
Rank : 5
Schlafli Type : {6,6,2,2}
Number of vertices, edges, etc : 18, 54, 18, 2, 2
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,2,2,2} of size 1728
Vertex Figure Of :
   {2,6,6,2,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6,2,2}*288a, {6,6,2,2}*288b, {6,6,2,2}*288c
   6-fold quotients : {3,6,2,2}*144, {6,3,2,2}*144
   9-fold quotients : {2,6,2,2}*96, {6,2,2,2}*96
   18-fold quotients : {2,3,2,2}*48, {3,2,2,2}*48
   27-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6,2,4}*1728d, {6,12,2,2}*1728g, {12,6,2,2}*1728g, {6,6,4,2}*1728h
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)
(17,23)(18,24);;
s1 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)
(20,24)(21,23)(26,27);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)
(17,25)(18,27);;
s3 := (28,29);;
s4 := (30,31);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, 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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(31)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)
(16,22)(17,23)(18,24);
s1 := Sym(31)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)
(19,22)(20,24)(21,23)(26,27);
s2 := Sym(31)!( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)
(16,26)(17,25)(18,27);
s3 := Sym(31)!(28,29);
s4 := Sym(31)!(30,31);
poly := sub<Sym(31)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;

to this polytope