Polytope of Type {3,2,4,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,28}*1344
if this polytope has a name.
Group : SmallGroup(1344,7765)
Rank : 5
Schlafli Type : {3,2,4,28}
Number of vertices, edges, etc : 3, 3, 4, 56, 28
Order of s₀s₁s₂s₃s₄ : 84
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 : {3,2,2,28}*672, {3,2,4,14}*672
   4-fold quotients : {3,2,2,14}*336
   7-fold quotients : {3,2,4,4}*192
   8-fold quotients : {3,2,2,7}*168
   14-fold quotients : {3,2,2,4}*96, {3,2,4,2}*96
   28-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

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