Polytope of Type {20,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4}*1000
if this polytope has a name.
Group : SmallGroup(1000,92)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 125, 250, 25
Order of s₀s₁s₂ : 10
Order of s₀s₁s₂s₁ : 5
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {20,4,2} of size 2000
Vertex Figure Of :
   {2,20,4} of size 2000
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {4,4}*200
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,4}*2000a
Permutation Representation (GAP) :

s0 := (  2,  5)(  3,  4)(  6, 32)(  7, 31)(  8, 35)(  9, 34)( 10, 33)( 11, 64)
( 12, 63)( 13, 62)( 14, 61)( 15, 65)( 16, 92)( 17, 91)( 18, 95)( 19, 94)
( 20, 93)( 21,121)( 22,125)( 23,124)( 24,123)( 25,122)( 26,101)( 27,105)
( 28,104)( 29,103)( 30,102)( 36, 39)( 37, 38)( 41, 67)( 42, 66)( 43, 70)
( 44, 69)( 45, 68)( 46, 96)( 47,100)( 48, 99)( 49, 98)( 50, 97)( 51, 76)
( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56,107)( 57,106)( 58,110)( 59,109)
( 60,108)( 72, 75)( 73, 74)( 81, 82)( 83, 85)( 86,114)( 87,113)( 88,112)
( 89,111)( 90,115)(116,117)(118,120);;
s1 := (  1,  2)(  3,  5)(  6, 95)(  7, 94)(  8, 93)(  9, 92)( 10, 91)( 11, 32)
( 12, 31)( 13, 35)( 14, 34)( 15, 33)( 16,123)( 17,122)( 18,121)( 19,125)
( 20,124)( 21, 63)( 22, 62)( 23, 61)( 24, 65)( 25, 64)( 26, 72)( 27, 71)
( 28, 75)( 29, 74)( 30, 73)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)
( 41, 44)( 42, 43)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 51,120)
( 52,119)( 53,118)( 54,117)( 55,116)( 56, 57)( 58, 60)( 66, 88)( 67, 87)
( 68, 86)( 69, 90)( 70, 89)( 81,105)( 82,104)( 83,103)( 84,102)( 85,101)
( 97,100)( 98, 99)(112,115)(113,114);;
s2 := (  1, 40)(  2, 39)(  3, 38)(  4, 37)(  5, 36)(  6,  7)(  8, 10)( 11,105)
( 12,104)( 13,103)( 14,102)( 15,101)( 16, 99)( 17, 98)( 18, 97)( 19, 96)
( 20,100)( 21, 69)( 22, 68)( 23, 67)( 24, 66)( 25, 70)( 26, 65)( 27, 64)
( 28, 63)( 29, 62)( 30, 61)( 31, 32)( 33, 35)( 41,124)( 42,123)( 43,122)
( 44,121)( 45,125)( 46, 94)( 47, 93)( 48, 92)( 49, 91)( 50, 95)( 51, 90)
( 52, 89)( 53, 88)( 54, 87)( 55, 86)( 56, 57)( 58, 60)( 71,119)( 72,118)
( 73,117)( 74,116)( 75,120)( 76,115)( 77,114)( 78,113)( 79,112)( 80,111)
( 81, 82)( 83, 85)(106,107)(108,110);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(125)!(  2,  5)(  3,  4)(  6, 32)(  7, 31)(  8, 35)(  9, 34)( 10, 33)
( 11, 64)( 12, 63)( 13, 62)( 14, 61)( 15, 65)( 16, 92)( 17, 91)( 18, 95)
( 19, 94)( 20, 93)( 21,121)( 22,125)( 23,124)( 24,123)( 25,122)( 26,101)
( 27,105)( 28,104)( 29,103)( 30,102)( 36, 39)( 37, 38)( 41, 67)( 42, 66)
( 43, 70)( 44, 69)( 45, 68)( 46, 96)( 47,100)( 48, 99)( 49, 98)( 50, 97)
( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56,107)( 57,106)( 58,110)
( 59,109)( 60,108)( 72, 75)( 73, 74)( 81, 82)( 83, 85)( 86,114)( 87,113)
( 88,112)( 89,111)( 90,115)(116,117)(118,120);
s1 := Sym(125)!(  1,  2)(  3,  5)(  6, 95)(  7, 94)(  8, 93)(  9, 92)( 10, 91)
( 11, 32)( 12, 31)( 13, 35)( 14, 34)( 15, 33)( 16,123)( 17,122)( 18,121)
( 19,125)( 20,124)( 21, 63)( 22, 62)( 23, 61)( 24, 65)( 25, 64)( 26, 72)
( 27, 71)( 28, 75)( 29, 74)( 30, 73)( 36, 76)( 37, 80)( 38, 79)( 39, 78)
( 40, 77)( 41, 44)( 42, 43)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)
( 51,120)( 52,119)( 53,118)( 54,117)( 55,116)( 56, 57)( 58, 60)( 66, 88)
( 67, 87)( 68, 86)( 69, 90)( 70, 89)( 81,105)( 82,104)( 83,103)( 84,102)
( 85,101)( 97,100)( 98, 99)(112,115)(113,114);
s2 := Sym(125)!(  1, 40)(  2, 39)(  3, 38)(  4, 37)(  5, 36)(  6,  7)(  8, 10)
( 11,105)( 12,104)( 13,103)( 14,102)( 15,101)( 16, 99)( 17, 98)( 18, 97)
( 19, 96)( 20,100)( 21, 69)( 22, 68)( 23, 67)( 24, 66)( 25, 70)( 26, 65)
( 27, 64)( 28, 63)( 29, 62)( 30, 61)( 31, 32)( 33, 35)( 41,124)( 42,123)
( 43,122)( 44,121)( 45,125)( 46, 94)( 47, 93)( 48, 92)( 49, 91)( 50, 95)
( 51, 90)( 52, 89)( 53, 88)( 54, 87)( 55, 86)( 56, 57)( 58, 60)( 71,119)
( 72,118)( 73,117)( 74,116)( 75,120)( 76,115)( 77,114)( 78,113)( 79,112)
( 80,111)( 81, 82)( 83, 85)(106,107)(108,110);
poly := sub<Sym(125)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
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