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}*1296o
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}*648c
   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*s1*s2*s1*s0*s2*s1*s2*s1*s2> 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, 60)( 11, 59)( 12, 58)( 13, 57)( 14, 56)( 15, 55)( 16, 62)( 17, 61)( 18, 63)( 19, 32)( 20, 31)( 21, 33)( 22, 29)( 23, 28)( 24, 30)( 25, 34)( 26, 36)( 27, 35)( 37, 77)( 38, 76)( 39, 78)( 40, 74)( 41, 73)( 42, 75)( 43, 79)( 44, 81)( 45, 80)( 46, 50)( 47, 49)( 48, 51)( 53, 54)( 64, 67)( 65, 69)( 66, 68)( 70, 72)( 82, 85)( 83, 87)( 84, 86)( 88, 90)( 91,141)( 92,140)( 93,139)( 94,138)( 95,137)( 96,136)( 97,143)( 98,142)( 99,144)(100,113)(101,112)(102,114)(103,110)(104,109)(105,111)(106,115)(107,117)(108,116)(118,158)(119,157)(120,159)(121,155)(122,154)(123,156)(124,160)(125,162)(126,161)(127,131)(128,130)(129,132)(134,135)(145,148)(146,150)(147,149)(151,153);;
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, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s0 ];;
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, 60)( 11, 59)( 12, 58)( 13, 57)( 14, 56)( 15, 55)( 16, 62)( 17, 61)( 18, 63)( 19, 32)( 20, 31)( 21, 33)( 22, 29)( 23, 28)( 24, 30)( 25, 34)( 26, 36)( 27, 35)( 37, 77)( 38, 76)( 39, 78)( 40, 74)( 41, 73)( 42, 75)( 43, 79)( 44, 81)( 45, 80)( 46, 50)( 47, 49)( 48, 51)( 53, 54)( 64, 67)( 65, 69)( 66, 68)( 70, 72)( 82, 85)( 83, 87)( 84, 86)( 88, 90)( 91,141)( 92,140)( 93,139)( 94,138)( 95,137)( 96,136)( 97,143)( 98,142)( 99,144)(100,113)(101,112)(102,114)(103,110)(104,109)(105,111)(106,115)(107,117)(108,116)(118,158)(119,157)(120,159)(121,155)(122,154)(123,156)(124,160)(125,162)(126,161)(127,131)(128,130)(129,132)(134,135)(145,148)(146,150)(147,149)(151,153);
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, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s0 >;
References : None.
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