Polytope of Type {36,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296n
if this polytope has a name.
Group : SmallGroup(1296,2978)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
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 : {36,6}*648b
   3-fold quotients : {12,6}*432i
   6-fold quotients : {12,6}*216c
   9-fold quotients : {4,6}*144
   18-fold quotients : {4,6}*72
   27-fold quotients : {12,2}*48
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {3,2}*12
   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*s2*s1*s0*s2*s1*s2*s1> of order 3.
      6 facets:
         6 of {36}*72
      36 vertex figures:
         36 of {6}*12

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