Polytope of Type {40,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40,4}*1280b
if this polytope has a name.
Group : SmallGroup(1280,90280)
Rank : 3
Schlafli Type : {40,4}
Number of vertices, edges, etc : 160, 320, 16
Order of s0s1s2 : 20
Order of s0s1s2s1 : 8
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) :
   2-fold quotients : {40,4}*640b
   4-fold quotients : {20,4}*320
   5-fold quotients : {8,4}*256b
   8-fold quotients : {20,4}*160
   10-fold quotients : {8,4}*128b
   16-fold quotients : {20,2}*80, {10,4}*80
   20-fold quotients : {4,4}*64
   32-fold quotients : {10,2}*40
   40-fold quotients : {4,4}*32
   64-fold quotients : {5,2}*20
   80-fold quotients : {2,4}*16, {4,2}*16
   160-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*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      12 facets:
         8 of {20}*40
         4 of {40}*80
      80 vertex figures:
         80 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 4.
      6 facets:
         4 of {20}*40
         2 of {40}*80
      40 vertex figures:
         40 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      10 facets:
         8 of {10}*20
         2 of {40}*80
      40 vertex figures:
         40 of {4}*8

Permutation Representation (GAP) : 
s0 := (  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)(  8, 89)(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)( 16, 96)( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,106)( 22,110)( 23,109)( 24,108)( 25,107)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,116)( 32,120)( 33,119)( 34,118)( 35,117)( 36,111)( 37,115)( 38,114)( 39,113)( 40,112)( 41,156)( 42,160)( 43,159)( 44,158)( 45,157)( 46,151)( 47,155)( 48,154)( 49,153)( 50,152)( 51,146)( 52,150)( 53,149)( 54,148)( 55,147)( 56,141)( 57,145)( 58,144)( 59,143)( 60,142)( 61,136)( 62,140)( 63,139)( 64,138)( 65,137)( 66,131)( 67,135)( 68,134)( 69,133)( 70,132)( 71,126)( 72,130)( 73,129)( 74,128)( 75,127)( 76,121)( 77,125)( 78,124)( 79,123)( 80,122);;
s1 := (  1,  3)(  4,  5)(  6,  8)(  9, 10)( 11, 13)( 14, 15)( 16, 18)( 19, 20)( 21, 33)( 22, 32)( 23, 31)( 24, 35)( 25, 34)( 26, 38)( 27, 37)( 28, 36)( 29, 40)( 30, 39)( 41, 43)( 44, 45)( 46, 48)( 49, 50)( 51, 53)( 54, 55)( 56, 58)( 59, 60)( 61, 73)( 62, 72)( 63, 71)( 64, 75)( 65, 74)( 66, 78)( 67, 77)( 68, 76)( 69, 80)( 70, 79)( 81,123)( 82,122)( 83,121)( 84,125)( 85,124)( 86,128)( 87,127)( 88,126)( 89,130)( 90,129)( 91,133)( 92,132)( 93,131)( 94,135)( 95,134)( 96,138)( 97,137)( 98,136)( 99,140)(100,139)(101,153)(102,152)(103,151)(104,155)(105,154)(106,158)(107,157)(108,156)(109,160)(110,159)(111,143)(112,142)(113,141)(114,145)(115,144)(116,148)(117,147)(118,146)(119,150)(120,149);;
s2 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 41, 76)( 42, 77)( 43, 78)( 44, 79)( 45, 80)( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 66)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 91, 96)( 92, 97)( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)(115,120)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150);;
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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(160)!(  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)(  8, 89)(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)( 16, 96)( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,106)( 22,110)( 23,109)( 24,108)( 25,107)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,116)( 32,120)( 33,119)( 34,118)( 35,117)( 36,111)( 37,115)( 38,114)( 39,113)( 40,112)( 41,156)( 42,160)( 43,159)( 44,158)( 45,157)( 46,151)( 47,155)( 48,154)( 49,153)( 50,152)( 51,146)( 52,150)( 53,149)( 54,148)( 55,147)( 56,141)( 57,145)( 58,144)( 59,143)( 60,142)( 61,136)( 62,140)( 63,139)( 64,138)( 65,137)( 66,131)( 67,135)( 68,134)( 69,133)( 70,132)( 71,126)( 72,130)( 73,129)( 74,128)( 75,127)( 76,121)( 77,125)( 78,124)( 79,123)( 80,122);
s1 := Sym(160)!(  1,  3)(  4,  5)(  6,  8)(  9, 10)( 11, 13)( 14, 15)( 16, 18)( 19, 20)( 21, 33)( 22, 32)( 23, 31)( 24, 35)( 25, 34)( 26, 38)( 27, 37)( 28, 36)( 29, 40)( 30, 39)( 41, 43)( 44, 45)( 46, 48)( 49, 50)( 51, 53)( 54, 55)( 56, 58)( 59, 60)( 61, 73)( 62, 72)( 63, 71)( 64, 75)( 65, 74)( 66, 78)( 67, 77)( 68, 76)( 69, 80)( 70, 79)( 81,123)( 82,122)( 83,121)( 84,125)( 85,124)( 86,128)( 87,127)( 88,126)( 89,130)( 90,129)( 91,133)( 92,132)( 93,131)( 94,135)( 95,134)( 96,138)( 97,137)( 98,136)( 99,140)(100,139)(101,153)(102,152)(103,151)(104,155)(105,154)(106,158)(107,157)(108,156)(109,160)(110,159)(111,143)(112,142)(113,141)(114,145)(115,144)(116,148)(117,147)(118,146)(119,150)(120,149);
s2 := Sym(160)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 41, 76)( 42, 77)( 43, 78)( 44, 79)( 45, 80)( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 66)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 91, 96)( 92, 97)( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)(115,120)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150);
poly := sub<Sym(160)|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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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