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}*1280c
if this polytope has a name.
Group : SmallGroup(1280,1116459)
Rank : 3
Schlafli Type : {4,10}
Number of vertices, edges, etc : 64, 320, 160
Order of s0s1s2 : 10
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      80 facets:
         80 of {4}*8
      32 vertex figures:
         32 of {10}*20
   P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      80 facets:
         80 of {4}*8
      32 vertex figures:
         32 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*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=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      80 facets:
         80 of {4}*8
      48 vertex figures:
         32 of {5}*10
         16 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*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*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*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s2*s1*s0*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
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*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=<s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1, s1*s0*s2*s1*s0*s2*s1*s0*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, 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);;
s1 := (  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)( 17, 90)( 18, 89)( 19, 92)( 20, 91)( 21, 94)( 22, 93)( 23, 96)( 24, 95)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)( 41, 49)( 42, 50)( 43, 51)( 44, 52)( 45, 53)( 46, 54)( 47, 55)( 48, 56)( 57,105)( 58,106)( 59,107)( 60,108)( 61,109)( 62,110)( 63,111)( 64,112)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 97,122)( 98,121)( 99,124)(100,123)(101,126)(102,125)(103,128)(104,127);;
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, 
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*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := 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);
s1 := Sym(128)!(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)( 17, 90)( 18, 89)( 19, 92)( 20, 91)( 21, 94)( 22, 93)( 23, 96)( 24, 95)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)( 41, 49)( 42, 50)( 43, 51)( 44, 52)( 45, 53)( 46, 54)( 47, 55)( 48, 56)( 57,105)( 58,106)( 59,107)( 60,108)( 61,109)( 62,110)( 63,111)( 64,112)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 97,122)( 98,121)( 99,124)(100,123)(101,126)(102,125)(103,128)(104,127);
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, 
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*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 >;
References : None.
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