Polytope of Type {10,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,8}*1280f
if this polytope has a name.
Group : SmallGroup(1280,1116459)
Rank : 3
Schlafli Type : {10,8}
Number of vertices, edges, etc : 80, 320, 64
Order of s0s1s2 : 10
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {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*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      32 facets:
         32 of {10}*20
      40 vertex figures:
         40 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 2.
      32 facets:
         32 of {10}*20
      40 vertex figures:
         40 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      32 facets:
         32 of {10}*20
      40 vertex figures:
         40 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2> of order 2.
      32 facets:
         32 of {10}*20
      40 vertex figures:
         40 of {8}*16
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1> of order 2.
      32 facets:
         32 of {10}*20
      40 vertex figures:
         40 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1> of order 4.
      16 facets:
         16 of {10}*20
      20 vertex figures:
         20 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
      16 facets:
         16 of {10}*20
      20 vertex figures:
         20 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
      16 facets:
         16 of {10}*20
      20 vertex figures:
         20 of {8}*16

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