Polytope of Type {4,40}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,40}*1280e
if this polytope has a name.
Group : SmallGroup(1280,1116431)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 16, 320, 160
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
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 : {4,20}*640b
   4-fold quotients : {4,10}*320a
   8-fold quotients : {4,5}*160
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*s2*s1*s0*s1*s2*s1> of order 2.
      96 facets:
         64 of {4}*8
         32 of {2}*4
      8 vertex figures:
         8 of {40}*80
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      80 facets:
         80 of {4}*8
      8 vertex figures:
         8 of {40}*80
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      80 facets:
         80 of {4}*8
      8 vertex figures:
         8 of {40}*80
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      56 facets:
         24 of {4}*8
         32 of {2}*4
      4 vertex figures:
         4 of {40}*80
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      48 facets:
         32 of {4}*8
         16 of {2}*4
      4 vertex figures:
         4 of {40}*80
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      48 facets:
         32 of {4}*8
         16 of {2}*4
      4 vertex figures:
         4 of {40}*80

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