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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,2,6,18}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {4,2,6,18}
Number of vertices, edges, etc : 4, 4, 6, 54, 18
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 : {2,2,6,18}*864a
   3-fold quotients : {4,2,2,18}*576, {4,2,6,6}*576a
   6-fold quotients : {4,2,2,9}*288, {2,2,2,18}*288, {2,2,6,6}*288a
   9-fold quotients : {4,2,2,6}*192, {4,2,6,2}*192
   12-fold quotients : {2,2,2,9}*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 := ( 8,11)( 9,12)(10,13)(17,20)(18,21)(19,22)(26,29)(27,30)(28,31)(35,38)
(36,39)(37,40)(44,47)(45,48)(46,49)(53,56)(54,57)(55,58);;
s3 := ( 5, 8)( 6,10)( 7, 9)(12,13)(14,27)(15,26)(16,28)(17,24)(18,23)(19,25)
(20,30)(21,29)(22,31)(32,35)(33,37)(34,36)(39,40)(41,54)(42,53)(43,55)(44,51)
(45,50)(46,52)(47,57)(48,56)(49,58);;
s4 := ( 5,41)( 6,43)( 7,42)( 8,44)( 9,46)(10,45)(11,47)(12,49)(13,48)(14,32)
(15,34)(16,33)(17,35)(18,37)(19,36)(20,38)(21,40)(22,39)(23,51)(24,50)(25,52)
(26,54)(27,53)(28,55)(29,57)(30,56)(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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope