Polytope of Type {4,2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,2,18,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {4,2,18,6}
Number of vertices, edges, etc : 4, 4, 18, 54, 6
Order of s₀s₁s₂s₃s₄ : 36
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,18,6}*864a
   3-fold quotients : {4,2,18,2}*576, {4,2,6,6}*576a
   6-fold quotients : {4,2,9,2}*288, {2,2,18,2}*288, {2,2,6,6}*288a
   9-fold quotients : {4,2,2,6}*192, {4,2,6,2}*192
   12-fold quotients : {2,2,9,2}*144
   18-fold quotients : {4,2,2,3}*96, {4,2,3,2}*96, {2,2,2,6}*96, {2,2,6,2}*96
   27-fold quotients : {4,2,2,2}*64
   36-fold quotients : {2,2,2,3}*48, {2,2,3,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := ( 6, 7)( 9,10)(12,13)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)(20,30)
(21,29)(22,31)(33,34)(36,37)(39,40)(41,51)(42,50)(43,52)(44,54)(45,53)(46,55)
(47,57)(48,56)(49,58);;
s3 := ( 5,14)( 6,16)( 7,15)( 8,20)( 9,22)(10,21)(11,17)(12,19)(13,18)(23,24)
(26,30)(27,29)(28,31)(32,41)(33,43)(34,42)(35,47)(36,49)(37,48)(38,44)(39,46)
(40,45)(50,51)(53,57)(54,56)(55,58);;
s4 := ( 5,35)( 6,36)( 7,37)( 8,32)( 9,33)(10,34)(11,38)(12,39)(13,40)(14,44)
(15,45)(16,46)(17,41)(18,42)(19,43)(20,47)(21,48)(22,49)(23,53)(24,54)(25,55)
(26,50)(27,51)(28,52)(29,56)(30,57)(31,58);;
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*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*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(58)!(2,3);
s1 := Sym(58)!(1,2)(3,4);
s2 := Sym(58)!( 6, 7)( 9,10)(12,13)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)
(20,30)(21,29)(22,31)(33,34)(36,37)(39,40)(41,51)(42,50)(43,52)(44,54)(45,53)
(46,55)(47,57)(48,56)(49,58);
s3 := Sym(58)!( 5,14)( 6,16)( 7,15)( 8,20)( 9,22)(10,21)(11,17)(12,19)(13,18)
(23,24)(26,30)(27,29)(28,31)(32,41)(33,43)(34,42)(35,47)(36,49)(37,48)(38,44)
(39,46)(40,45)(50,51)(53,57)(54,56)(55,58);
s4 := Sym(58)!( 5,35)( 6,36)( 7,37)( 8,32)( 9,33)(10,34)(11,38)(12,39)(13,40)
(14,44)(15,45)(16,46)(17,41)(18,42)(19,43)(20,47)(21,48)(22,49)(23,53)(24,54)
(25,55)(26,50)(27,51)(28,52)(29,56)(30,57)(31,58);
poly := sub<Sym(58)|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*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope