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}*1280d
if this polytope has a name.
Group : SmallGroup(1280,1116454)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 160, 320, 32
Order of s0s1s2 : 10
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 : {20,4}*640d, {20,4}*640e, {10,4}*640b
   4-fold quotients : {5,4}*320, {10,4}*320a, {10,4}*320b
   8-fold quotients : {5,4}*160
   32-fold quotients : {10,2}*40
   64-fold quotients : {5,2}*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*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1> of order 2.
      16 facets:
         16 of {20}*40
      80 vertex figures:
         80 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 2.
      16 facets:
         16 of {20}*40
      80 vertex figures:
         80 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      16 facets:
         16 of {20}*40
      80 vertex figures:
         80 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2> of order 2.
      16 facets:
         16 of {20}*40
      80 vertex figures:
         80 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {20}*40
      40 vertex figures:
         40 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      8 facets:
         8 of {20}*40
      40 vertex figures:
         40 of {4}*8

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