Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1296e
if this polytope has a name.
Group : SmallGroup(1296,2977)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 54, 324, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
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 : {12,12}*648
   3-fold quotients : {4,12}*432a, {12,4}*432a
   6-fold quotients : {4,12}*216, {12,4}*216
   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=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {6}*12
         24 of {12}*24
      30 vertex figures:
         24 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      15 facets:
         3 of {6}*12
         12 of {12}*24
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12

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