Polytope of Type {4,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,10}*1280b
if this polytope has a name.
Group : SmallGroup(1280,1116454)
Rank : 3
Schlafli Type : {4,10}
Number of vertices, edges, etc : 64, 320, 160
Order of s0s1s2 : 20
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 : {4,5}*640, {4,10}*640a, {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=<s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      88 facets:
         72 of {4}*8
         16 of {2}*4
      32 vertex figures:
         32 of {10}*20
   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
      32 vertex figures:
         32 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 2.
      80 facets:
         80 of {4}*8
      32 vertex figures:
         32 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      44 facets:
         36 of {4}*8
         8 of {2}*4
      16 vertex figures:
         16 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1> of order 4.
      44 facets:
         36 of {4}*8
         8 of {2}*4
      16 vertex figures:
         16 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      44 facets:
         36 of {4}*8
         8 of {2}*4
      16 vertex figures:
         16 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 4.
      44 facets:
         36 of {4}*8
         8 of {2}*4
      16 vertex figures:
         16 of {10}*20

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