Polytope of Type {6,36}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1296o
if this polytope has a name.
Group : SmallGroup(1296,2978)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,36}*648c
   3-fold quotients : {6,12}*432i
   6-fold quotients : {6,12}*216c
   9-fold quotients : {6,4}*144
   18-fold quotients : {6,4}*72
   27-fold quotients : {2,12}*48
   54-fold quotients : {2,6}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {2,3}*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*s0*s1*s2*s1*s0*s1*s0*s2> of order 3.
      36 facets:
         36 of {6}*12
      6 vertex figures:
         6 of {36}*72

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