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}*1728u
if this polytope has a name.
Group : SmallGroup(1728,46099)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 3
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   3-fold quotients : {12,12}*576l
   4-fold quotients : {6,12}*432d, {12,6}*432d
   12-fold quotients : {6,12}*144d, {12,6}*144d
   16-fold quotients : {6,6}*108
   36-fold quotients : {4,6}*48b, {6,4}*48b
   72-fold quotients : {3,4}*24, {4,3}*24
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*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      48 vertex figures:
         24 of {12}*24
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      48 facets:
         24 of {6}*12
         24 of {12}*24
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      24 facets:
         12 of {6}*12
         12 of {12}*24
      24 vertex figures:
         12 of {12}*24
         12 of {6}*12

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