Polytope of Type {8,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,10}*1280c
if this polytope has a name.
Group : SmallGroup(1280,1116450)
Rank : 3
Schlafli Type : {8,10}
Number of vertices, edges, etc : 64, 320, 80
Order of s0s1s2 : 10
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {8,5}*640a, {8,10}*640a, {8,10}*640b, {8,10}*640c, {8,10}*640d, {4,10}*640b
   4-fold quotients : {8,5}*320a, {8,5}*320b, {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*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 2.
      40 facets:
         40 of {8}*16
      32 vertex figures:
         32 of {10}*20
   P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      40 facets:
         40 of {8}*16
      32 vertex figures:
         32 of {10}*20
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      20 facets:
         20 of {8}*16
      16 vertex figures:
         16 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      20 facets:
         20 of {8}*16
      16 vertex figures:
         16 of {10}*20

Permutation Representation (GAP) : 
s0 := (  1,101)(  2,102)(  3,103)(  4,104)(  5, 97)(  6, 98)(  7, 99)(  8,100)(  9,110)( 10,109)( 11,112)( 12,111)( 13,106)( 14,105)( 15,108)( 16,107)( 17,117)( 18,118)( 19,119)( 20,120)( 21,113)( 22,114)( 23,115)( 24,116)( 25,126)( 26,125)( 27,128)( 28,127)( 29,122)( 30,121)( 31,124)( 32,123)( 33, 69)( 34, 70)( 35, 71)( 36, 72)( 37, 65)( 38, 66)( 39, 67)( 40, 68)( 41, 78)( 42, 77)( 43, 80)( 44, 79)( 45, 74)( 46, 73)( 47, 76)( 48, 75)( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 81)( 54, 82)( 55, 83)( 56, 84)( 57, 94)( 58, 93)( 59, 96)( 60, 95)( 61, 90)( 62, 89)( 63, 92)( 64, 91);;
s1 := ( 17, 26)( 18, 25)( 19, 28)( 20, 27)( 21, 30)( 22, 29)( 23, 32)( 24, 31)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65,121)( 66,122)( 67,123)( 68,124)( 69,125)( 70,126)( 71,127)( 72,128)( 73,113)( 74,114)( 75,115)( 76,116)( 77,117)( 78,118)( 79,119)( 80,120)( 81, 97)( 82, 98)( 83, 99)( 84,100)( 85,101)( 86,102)( 87,103)( 88,104)( 89,105)( 90,106)( 91,107)( 92,108)( 93,109)( 94,110)( 95,111)( 96,112);;
s2 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9,123)( 10,124)( 11,121)( 12,122)( 13,127)( 14,128)( 15,125)( 16,126)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25,107)( 26,108)( 27,105)( 28,106)( 29,111)( 30,112)( 31,109)( 32,110)( 33, 52)( 34, 51)( 35, 50)( 36, 49)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 75)( 42, 76)( 43, 73)( 44, 74)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 57, 92)( 58, 91)( 59, 90)( 60, 89)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 65, 84)( 66, 83)( 67, 82)( 68, 81)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 97, 99)( 98,100)(101,103)(102,104)(113,116)(114,115)(117,120)(118,119);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1, 
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)!(  1,101)(  2,102)(  3,103)(  4,104)(  5, 97)(  6, 98)(  7, 99)(  8,100)(  9,110)( 10,109)( 11,112)( 12,111)( 13,106)( 14,105)( 15,108)( 16,107)( 17,117)( 18,118)( 19,119)( 20,120)( 21,113)( 22,114)( 23,115)( 24,116)( 25,126)( 26,125)( 27,128)( 28,127)( 29,122)( 30,121)( 31,124)( 32,123)( 33, 69)( 34, 70)( 35, 71)( 36, 72)( 37, 65)( 38, 66)( 39, 67)( 40, 68)( 41, 78)( 42, 77)( 43, 80)( 44, 79)( 45, 74)( 46, 73)( 47, 76)( 48, 75)( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 81)( 54, 82)( 55, 83)( 56, 84)( 57, 94)( 58, 93)( 59, 96)( 60, 95)( 61, 90)( 62, 89)( 63, 92)( 64, 91);
s1 := Sym(128)!( 17, 26)( 18, 25)( 19, 28)( 20, 27)( 21, 30)( 22, 29)( 23, 32)( 24, 31)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65,121)( 66,122)( 67,123)( 68,124)( 69,125)( 70,126)( 71,127)( 72,128)( 73,113)( 74,114)( 75,115)( 76,116)( 77,117)( 78,118)( 79,119)( 80,120)( 81, 97)( 82, 98)( 83, 99)( 84,100)( 85,101)( 86,102)( 87,103)( 88,104)( 89,105)( 90,106)( 91,107)( 92,108)( 93,109)( 94,110)( 95,111)( 96,112);
s2 := Sym(128)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9,123)( 10,124)( 11,121)( 12,122)( 13,127)( 14,128)( 15,125)( 16,126)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25,107)( 26,108)( 27,105)( 28,106)( 29,111)( 30,112)( 31,109)( 32,110)( 33, 52)( 34, 51)( 35, 50)( 36, 49)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 75)( 42, 76)( 43, 73)( 44, 74)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 57, 92)( 58, 91)( 59, 90)( 60, 89)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 65, 84)( 66, 83)( 67, 82)( 68, 81)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 97, 99)( 98,100)(101,103)(102,104)(113,116)(114,115)(117,120)(118,119);
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1, 
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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