Polytope of Type {20,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4}*2000b
if this polytope has a name.
Group : SmallGroup(2000,919)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 250, 500, 50
Order of s0s1s2 : 10
Order of s0s1s2s1 : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
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 : {10,2}*40
   100-fold quotients : {5,2}*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*s1*s2*s1*s0*s1*s2*s1> of order 2.
      25 facets:
         25 of {20}*40
      125 vertex figures:
         125 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      25 facets:
         25 of {20}*40
      125 vertex figures:
         125 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      26 facets:
         2 of {10}*20
         24 of {20}*40
      130 vertex figures:
         120 of {4}*8
         10 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      13 facets:
         1 of {10}*20
         12 of {20}*40
      65 vertex figures:
         60 of {4}*8
         5 of {2}*4
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 5.
      10 facets:
         10 of {20}*40
      50 vertex figures:
         50 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 5.
      10 facets:
         10 of {20}*40
      50 vertex figures:
         50 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 5.
      10 facets:
         10 of {20}*40
      50 vertex figures:
         50 of {4}*8
   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.
      10 facets:
         10 of {20}*40
      50 vertex figures:
         50 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 5.
      10 facets:
         10 of {20}*40
      50 vertex figures:
         50 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 10.
      6 facets:
         2 of {10}*20
         4 of {20}*40
      30 vertex figures:
         20 of {4}*8
         10 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 10.
      5 facets:
         5 of {20}*40
      25 vertex figures:
         25 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 10.
      6 facets:
         2 of {10}*20
         4 of {20}*40
      30 vertex figures:
         20 of {4}*8
         10 of {2}*4

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