Polytope of Type {2,6,9,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9,2}*432
if this polytope has a name.
Group : SmallGroup(432,544)
Rank : 5
Schlafli Type : {2,6,9,2}
Number of vertices, edges, etc : 2, 6, 27, 9, 2
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,9,2,2} of size 864
   {2,6,9,2,3} of size 1296
   {2,6,9,2,4} of size 1728
Vertex Figure Of :
   {2,2,6,9,2} of size 864
   {3,2,6,9,2} of size 1296
   {4,2,6,9,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,9,2}*144, {2,6,3,2}*144
   9-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,9,2}*864, {2,6,18,2}*864b
   3-fold covers : {2,18,9,2}*1296, {2,6,9,2}*1296a, {2,6,27,2}*1296, {2,6,9,6}*1296, {6,6,9,2}*1296b
   4-fold covers : {8,6,9,2}*1728, {2,6,36,2}*1728b, {2,6,18,4}*1728b, {2,12,18,2}*1728b, {4,6,18,2}*1728b, {2,6,9,2}*1728, {2,6,9,4}*1728, {2,12,9,2}*1728
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

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