Polytope of Type {4,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*1296
Also Known As : {4,4}(9,9), {4,4}18. if this polytope has another name.
Group : SmallGroup(1296,1813)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 162, 324, 162
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*648
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72
   162-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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      81 facets:
         81 of {4}*8
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s0*s1> of order 2.
      82 facets:
         2 of {2}*4
         80 of {4}*8
      82 vertex figures:
         80 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      54 facets:
         54 of {4}*8
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {4}*8
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      41 facets:
         1 of {2}*4
         40 of {4}*8
      41 vertex figures:
         40 of {4}*8
         1 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      28 facets:
         2 of {2}*4
         26 of {4}*8
      28 vertex figures:
         26 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 6.
      28 facets:
         2 of {2}*4
         26 of {4}*8
      28 vertex figures:
         26 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 9.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 9.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 9.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 12.
      14 facets:
         1 of {2}*4
         13 of {4}*8
      14 vertex figures:
         13 of {4}*8
         1 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 18.
      10 facets:
         2 of {2}*4
         8 of {4}*8
      10 vertex figures:
         8 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 18.
      10 facets:
         2 of {2}*4
         8 of {4}*8
      10 vertex figures:
         8 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 18.
      10 facets:
         2 of {2}*4
         8 of {4}*8
      10 vertex figures:
         8 of {4}*8
         2 of {2}*4

Permutation Representation (GAP) : 
s0 := (  2,  8)(  3,  6)(  4,  7)( 10, 67)( 11, 65)( 12, 72)( 13, 64)( 14, 71)( 15, 69)( 16, 70)( 17, 68)( 18, 66)( 19, 46)( 20, 53)( 21, 51)( 22, 52)( 23, 50)( 24, 48)( 25, 49)( 26, 47)( 27, 54)( 28, 61)( 29, 59)( 30, 57)( 31, 58)( 32, 56)( 33, 63)( 34, 55)( 35, 62)( 36, 60)( 37, 40)( 39, 45)( 41, 44)( 73, 79)( 74, 77)( 78, 81)( 83, 89)( 84, 87)( 85, 88)( 91,148)( 92,146)( 93,153)( 94,145)( 95,152)( 96,150)( 97,151)( 98,149)( 99,147)(100,127)(101,134)(102,132)(103,133)(104,131)(105,129)(106,130)(107,128)(108,135)(109,142)(110,140)(111,138)(112,139)(113,137)(114,144)(115,136)(116,143)(117,141)(118,121)(120,126)(122,125)(154,160)(155,158)(159,162);;
s1 := (  2,  3)(  4,  5)(  7,  9)( 10, 11)( 13, 15)( 17, 18)( 19, 21)( 23, 24)( 25, 26)( 28, 80)( 29, 79)( 30, 81)( 31, 75)( 32, 74)( 33, 73)( 34, 76)( 35, 78)( 36, 77)( 37, 56)( 38, 55)( 39, 57)( 40, 60)( 41, 59)( 42, 58)( 43, 61)( 44, 63)( 45, 62)( 46, 66)( 47, 65)( 48, 64)( 49, 67)( 50, 69)( 51, 68)( 52, 71)( 53, 70)( 54, 72)( 83, 84)( 85, 86)( 88, 90)( 91, 92)( 94, 96)( 98, 99)(100,102)(104,105)(106,107)(109,161)(110,160)(111,162)(112,156)(113,155)(114,154)(115,157)(116,159)(117,158)(118,137)(119,136)(120,138)(121,141)(122,140)(123,139)(124,142)(125,144)(126,143)(127,147)(128,146)(129,145)(130,148)(131,150)(132,149)(133,152)(134,151)(135,153);;
s2 := (  1,119)(  2,121)(  3,126)(  4,122)(  5,124)(  6,120)(  7,125)(  8,118)(  9,123)( 10,140)( 11,142)( 12,138)( 13,143)( 14,136)( 15,141)( 16,137)( 17,139)( 18,144)( 19,108)( 20,101)( 21,103)( 22,102)( 23,104)( 24,106)( 25,105)( 26,107)( 27,100)( 28,146)( 29,148)( 30,153)( 31,149)( 32,151)( 33,147)( 34,152)( 35,145)( 36,150)( 37, 89)( 38, 82)( 39, 87)( 40, 83)( 41, 85)( 42, 90)( 43, 86)( 44, 88)( 45, 84)( 46,135)( 47,128)( 48,130)( 49,129)( 50,131)( 51,133)( 52,132)( 53,134)( 54,127)( 55, 95)( 56, 97)( 57, 93)( 58, 98)( 59, 91)( 60, 96)( 61, 92)( 62, 94)( 63, 99)( 64,116)( 65,109)( 66,114)( 67,110)( 68,112)( 69,117)( 70,113)( 71,115)( 72,111)( 73,162)( 74,155)( 75,157)( 76,156)( 77,158)( 78,160)( 79,159)( 80,161)( 81,154);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(162)!(  2,  8)(  3,  6)(  4,  7)( 10, 67)( 11, 65)( 12, 72)( 13, 64)( 14, 71)( 15, 69)( 16, 70)( 17, 68)( 18, 66)( 19, 46)( 20, 53)( 21, 51)( 22, 52)( 23, 50)( 24, 48)( 25, 49)( 26, 47)( 27, 54)( 28, 61)( 29, 59)( 30, 57)( 31, 58)( 32, 56)( 33, 63)( 34, 55)( 35, 62)( 36, 60)( 37, 40)( 39, 45)( 41, 44)( 73, 79)( 74, 77)( 78, 81)( 83, 89)( 84, 87)( 85, 88)( 91,148)( 92,146)( 93,153)( 94,145)( 95,152)( 96,150)( 97,151)( 98,149)( 99,147)(100,127)(101,134)(102,132)(103,133)(104,131)(105,129)(106,130)(107,128)(108,135)(109,142)(110,140)(111,138)(112,139)(113,137)(114,144)(115,136)(116,143)(117,141)(118,121)(120,126)(122,125)(154,160)(155,158)(159,162);
s1 := Sym(162)!(  2,  3)(  4,  5)(  7,  9)( 10, 11)( 13, 15)( 17, 18)( 19, 21)( 23, 24)( 25, 26)( 28, 80)( 29, 79)( 30, 81)( 31, 75)( 32, 74)( 33, 73)( 34, 76)( 35, 78)( 36, 77)( 37, 56)( 38, 55)( 39, 57)( 40, 60)( 41, 59)( 42, 58)( 43, 61)( 44, 63)( 45, 62)( 46, 66)( 47, 65)( 48, 64)( 49, 67)( 50, 69)( 51, 68)( 52, 71)( 53, 70)( 54, 72)( 83, 84)( 85, 86)( 88, 90)( 91, 92)( 94, 96)( 98, 99)(100,102)(104,105)(106,107)(109,161)(110,160)(111,162)(112,156)(113,155)(114,154)(115,157)(116,159)(117,158)(118,137)(119,136)(120,138)(121,141)(122,140)(123,139)(124,142)(125,144)(126,143)(127,147)(128,146)(129,145)(130,148)(131,150)(132,149)(133,152)(134,151)(135,153);
s2 := Sym(162)!(  1,119)(  2,121)(  3,126)(  4,122)(  5,124)(  6,120)(  7,125)(  8,118)(  9,123)( 10,140)( 11,142)( 12,138)( 13,143)( 14,136)( 15,141)( 16,137)( 17,139)( 18,144)( 19,108)( 20,101)( 21,103)( 22,102)( 23,104)( 24,106)( 25,105)( 26,107)( 27,100)( 28,146)( 29,148)( 30,153)( 31,149)( 32,151)( 33,147)( 34,152)( 35,145)( 36,150)( 37, 89)( 38, 82)( 39, 87)( 40, 83)( 41, 85)( 42, 90)( 43, 86)( 44, 88)( 45, 84)( 46,135)( 47,128)( 48,130)( 49,129)( 50,131)( 51,133)( 52,132)( 53,134)( 54,127)( 55, 95)( 56, 97)( 57, 93)( 58, 98)( 59, 91)( 60, 96)( 61, 92)( 62, 94)( 63, 99)( 64,116)( 65,109)( 66,114)( 67,110)( 68,112)( 69,117)( 70,113)( 71,115)( 72,111)( 73,162)( 74,155)( 75,157)( 76,156)( 77,158)( 78,160)( 79,159)( 80,161)( 81,154);
poly := sub<Sym(162)|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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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