Polytope of Type {3,2,9,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,9,4}*432
if this polytope has a name.
Group : SmallGroup(432,522)
Rank : 5
Schlafli Type : {3,2,9,4}
Number of vertices, edges, etc : 3, 3, 9, 18, 4
Order of s₀s₁s₂s₃s₄ : 9
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,9,4,2} of size 864
Vertex Figure Of :
   {2,3,2,9,4} of size 864
   {3,3,2,9,4} of size 1728
   {4,3,2,9,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,3,4}*144
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,9,4}*864, {3,2,18,4}*864b, {3,2,18,4}*864c, {6,2,9,4}*864
   3-fold covers : {3,2,27,4}*1296, {9,2,9,4}*1296, {3,6,9,4}*1296
   4-fold covers : {3,2,36,4}*1728b, {3,2,36,4}*1728c, {12,2,9,4}*1728, {3,2,9,8}*1728, {3,2,18,4}*1728, {6,2,9,4}*1728, {6,2,18,4}*1728b, {6,2,18,4}*1728c
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4, 5)( 6, 9)( 7, 8)(10,18)(11,17)(12,19)(13,15)(14,16)(20,26)(21,27)
(22,24)(23,25)(28,34)(29,35)(30,32)(31,33)(36,39)(37,38);;
s3 := ( 4, 8)( 5, 6)( 7,15)( 9,11)(10,12)(13,24)(14,25)(16,18)(17,20)(19,21)
(22,32)(23,33)(26,28)(27,29)(30,34)(31,38)(35,36)(37,39);;
s4 := ( 4,18)( 5,10)( 6,12)( 9,19)(13,23)(15,25)(20,29)(22,31)(24,33)(26,35)
(28,36)(34,39);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s3*s2*s4*s3*s4*s3*s2*s3, 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(39)!(2,3);
s1 := Sym(39)!(1,2);
s2 := Sym(39)!( 4, 5)( 6, 9)( 7, 8)(10,18)(11,17)(12,19)(13,15)(14,16)(20,26)
(21,27)(22,24)(23,25)(28,34)(29,35)(30,32)(31,33)(36,39)(37,38);
s3 := Sym(39)!( 4, 8)( 5, 6)( 7,15)( 9,11)(10,12)(13,24)(14,25)(16,18)(17,20)
(19,21)(22,32)(23,33)(26,28)(27,29)(30,34)(31,38)(35,36)(37,39);
s4 := Sym(39)!( 4,18)( 5,10)( 6,12)( 9,19)(13,23)(15,25)(20,29)(22,31)(24,33)
(26,35)(28,36)(34,39);
poly := sub<Sym(39)|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, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4, s4*s3*s2*s4*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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