Polytope of Type {2,4,2,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,2,54}*1728
if this polytope has a name.
Group : SmallGroup(1728,11400)
Rank : 5
Schlafli Type : {2,4,2,54}
Number of vertices, edges, etc : 2, 4, 4, 54, 54
Order of s₀s₁s₂s₃s₄ : 108
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 : {2,4,2,27}*864, {2,2,2,54}*864
   3-fold quotients : {2,4,2,18}*576
   4-fold quotients : {2,2,2,27}*432
   6-fold quotients : {2,4,2,9}*288, {2,2,2,18}*288
   9-fold quotients : {2,4,2,6}*192
   12-fold quotients : {2,2,2,9}*144
   18-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96
   27-fold quotients : {2,4,2,2}*64
   36-fold quotients : {2,2,2,3}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (4,5);;
s2 := (3,4)(5,6);;
s3 := ( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)
(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)
(51,52)(53,54)(55,56)(57,58)(59,60);;
s4 := ( 7,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,23)(20,21)(22,27)(24,25)
(26,31)(28,29)(30,35)(32,33)(34,39)(36,37)(38,43)(40,41)(42,47)(44,45)(46,51)
(48,49)(50,55)(52,53)(54,59)(56,57)(58,60);;
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, 
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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(60)!(1,2);
s1 := Sym(60)!(4,5);
s2 := Sym(60)!(3,4)(5,6);
s3 := Sym(60)!( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)
(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)
(49,50)(51,52)(53,54)(55,56)(57,58)(59,60);
s4 := Sym(60)!( 7,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,23)(20,21)(22,27)
(24,25)(26,31)(28,29)(30,35)(32,33)(34,39)(36,37)(38,43)(40,41)(42,47)(44,45)
(46,51)(48,49)(50,55)(52,53)(54,59)(56,57)(58,60);
poly := sub<Sym(60)|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, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2*s1*s2, 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*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*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