Polytope of Type {4,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*2000b
if this polytope has a name.
Group : SmallGroup(2000,919)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 50, 500, 250
Order of s0s1s2 : 10
Order of s0s1s2s1 : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {4,4}*400
   10-fold quotients : {4,4}*200
   50-fold quotients : {2,10}*40
   100-fold quotients : {2,5}*20
   250-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 2.
      125 facets:
         125 of {4}*8
      25 vertex figures:
         25 of {20}*40
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 2.
      125 facets:
         125 of {4}*8
      25 vertex figures:
         25 of {20}*40
   P/N, where N=<s0*s1*s0*s1> of order 2.
      130 facets:
         10 of {2}*4
         120 of {4}*8
      26 vertex figures:
         24 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      65 facets:
         5 of {2}*4
         60 of {4}*8
      13 vertex figures:
         12 of {20}*40
         1 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 10.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 10.
      25 facets:
         25 of {4}*8
      5 vertex figures:
         5 of {20}*40
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 10.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20

Permutation Representation (GAP) : 
s0 := (  6, 26)(  7, 27)(  8, 28)(  9, 29)( 10, 30)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21,101)( 22,102)( 23,103)( 24,104)( 25,105)( 36, 56)( 37, 57)( 38, 58)( 39, 59)( 40, 60)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 85)( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71,111)( 72,112)( 73,113)( 74,114)( 75,115)( 96,116)( 97,117)( 98,118)( 99,119)(100,120);;
s1 := (  2,  5)(  3,  4)(  6, 76)(  7, 80)(  8, 79)(  9, 78)( 10, 77)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 16,101)( 17,105)( 18,104)( 19,103)( 20,102)( 21, 51)( 22, 55)( 23, 54)( 24, 53)( 25, 52)( 31, 86)( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 37, 40)( 38, 39)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46, 61)( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 66,121)( 67,125)( 68,124)( 69,123)( 70,122)( 72, 75)( 73, 74)( 82, 85)( 83, 84)( 91,106)( 92,110)( 93,109)( 94,108)( 95,107)(117,120)(118,119);;
s2 := (  1, 32)(  2, 31)(  3, 35)(  4, 34)(  5, 33)(  6,  7)(  8, 10)( 11,107)( 12,106)( 13,110)( 14,109)( 15,108)( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 57)( 22, 56)( 23, 60)( 24, 59)( 25, 58)( 26, 27)( 28, 30)( 36,102)( 37,101)( 38,105)( 39,104)( 40,103)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 78)( 46, 52)( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)( 66, 97)( 67, 96)( 68,100)( 69, 99)( 70, 98)( 71, 72)( 73, 75)( 86,117)( 87,116)( 88,120)( 89,119)( 90,118)( 91, 92)( 93, 95)(111,112)(113,115);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(125)!(  6, 26)(  7, 27)(  8, 28)(  9, 29)( 10, 30)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21,101)( 22,102)( 23,103)( 24,104)( 25,105)( 36, 56)( 37, 57)( 38, 58)( 39, 59)( 40, 60)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 85)( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71,111)( 72,112)( 73,113)( 74,114)( 75,115)( 96,116)( 97,117)( 98,118)( 99,119)(100,120);
s1 := Sym(125)!(  2,  5)(  3,  4)(  6, 76)(  7, 80)(  8, 79)(  9, 78)( 10, 77)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 16,101)( 17,105)( 18,104)( 19,103)( 20,102)( 21, 51)( 22, 55)( 23, 54)( 24, 53)( 25, 52)( 31, 86)( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 37, 40)( 38, 39)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46, 61)( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 66,121)( 67,125)( 68,124)( 69,123)( 70,122)( 72, 75)( 73, 74)( 82, 85)( 83, 84)( 91,106)( 92,110)( 93,109)( 94,108)( 95,107)(117,120)(118,119);
s2 := Sym(125)!(  1, 32)(  2, 31)(  3, 35)(  4, 34)(  5, 33)(  6,  7)(  8, 10)( 11,107)( 12,106)( 13,110)( 14,109)( 15,108)( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 57)( 22, 56)( 23, 60)( 24, 59)( 25, 58)( 26, 27)( 28, 30)( 36,102)( 37,101)( 38,105)( 39,104)( 40,103)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 78)( 46, 52)( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)( 66, 97)( 67, 96)( 68,100)( 69, 99)( 70, 98)( 71, 72)( 73, 75)( 86,117)( 87,116)( 88,120)( 89,119)( 90,118)( 91, 92)( 93, 95)(111,112)(113,115);
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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