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}*1280e
if this polytope has a name.
Group : SmallGroup(1280,1116457)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 32, 320, 160
Order of s0s1s2 : 20
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {4,10}*640b
   4-fold quotients : {4,5}*320, {4,10}*320a, {4,10}*320b
   8-fold quotients : {4,5}*160
   32-fold quotients : {2,10}*40
   64-fold quotients : {2,5}*20
   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*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 2.
      80 facets:
         80 of {4}*8
      16 vertex figures:
         16 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      80 facets:
         80 of {4}*8
      16 vertex figures:
         16 of {20}*40
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
      80 facets:
         80 of {4}*8
      16 vertex figures:
         16 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 2.
      80 facets:
         80 of {4}*8
      16 vertex figures:
         16 of {20}*40
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
      40 facets:
         40 of {4}*8
      8 vertex figures:
         8 of {20}*40

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