Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*576b
if this polytope has a name.
Group : SmallGroup(576,8313)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 48, 144, 24
Order of s0s1s2 : 12
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 1152
Vertex Figure Of :
   {2,12,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6}*288a
   3-fold quotients : {4,6}*192b
   4-fold quotients : {12,6}*144a, {12,6}*144d
   6-fold quotients : {4,6}*96
   8-fold quotients : {6,6}*72a
   12-fold quotients : {12,2}*48, {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   24-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
   36-fold quotients : {4,2}*16
   48-fold quotients : {2,3}*12, {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6}*1152d, {24,6}*1152h, {12,6}*1152d, {12,12}*1152i, {12,12}*1152n
   3-fold covers : {36,6}*1728b, {12,18}*1728c, {12,6}*1728b, {12,6}*1728i
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      12 facets:
         12 of {12}*24
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      16 facets:
         8 of {12}*24
         8 of {6}*12
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      12 facets:
         12 of {12}*24
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s1*s2*s1*s2> of order 3.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         12 of {2}*4
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
      6 facets:
         6 of {12}*24
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      10 facets:
         8 of {6}*12
         2 of {12}*24
      12 vertex figures:
         12 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 15)( 14, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 27)( 26, 28)( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 49, 51)( 50, 52)( 53, 59)( 54, 60)( 55, 57)( 56, 58)( 61, 63)( 62, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73,111)( 74,112)( 75,109)( 76,110)( 77,119)( 78,120)( 79,117)( 80,118)( 81,115)( 82,116)( 83,113)( 84,114)( 85,123)( 86,124)( 87,121)( 88,122)( 89,131)( 90,132)( 91,129)( 92,130)( 93,127)( 94,128)( 95,125)( 96,126)( 97,135)( 98,136)( 99,133)(100,134)(101,143)(102,144)(103,141)(104,142)(105,139)(106,140)(107,137)(108,138);;
s1 := (  1, 77)(  2, 79)(  3, 78)(  4, 80)(  5, 73)(  6, 75)(  7, 74)(  8, 76)(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13,101)( 14,103)( 15,102)( 16,104)( 17, 97)( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)( 33, 93)( 34, 95)( 35, 94)( 36, 96)( 37,113)( 38,115)( 39,114)( 40,116)( 41,109)( 42,111)( 43,110)( 44,112)( 45,117)( 46,119)( 47,118)( 48,120)( 49,137)( 50,139)( 51,138)( 52,140)( 53,133)( 54,135)( 55,134)( 56,136)( 57,141)( 58,143)( 59,142)( 60,144)( 61,125)( 62,127)( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132);;
s2 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 29)(  6, 32)(  7, 31)(  8, 30)(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 14, 16)( 18, 20)( 22, 24)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 65)( 42, 68)( 43, 67)( 44, 66)( 45, 69)( 46, 72)( 47, 71)( 48, 70)( 50, 52)( 54, 56)( 58, 60)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,101)( 78,104)( 79,103)( 80,102)( 81,105)( 82,108)( 83,107)( 84,106)( 86, 88)( 90, 92)( 94, 96)(109,133)(110,136)(111,135)(112,134)(113,137)(114,140)(115,139)(116,138)(117,141)(118,144)(119,143)(120,142)(122,124)(126,128)(130,132);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*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(144)!(  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 15)( 14, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 27)( 26, 28)( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 49, 51)( 50, 52)( 53, 59)( 54, 60)( 55, 57)( 56, 58)( 61, 63)( 62, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73,111)( 74,112)( 75,109)( 76,110)( 77,119)( 78,120)( 79,117)( 80,118)( 81,115)( 82,116)( 83,113)( 84,114)( 85,123)( 86,124)( 87,121)( 88,122)( 89,131)( 90,132)( 91,129)( 92,130)( 93,127)( 94,128)( 95,125)( 96,126)( 97,135)( 98,136)( 99,133)(100,134)(101,143)(102,144)(103,141)(104,142)(105,139)(106,140)(107,137)(108,138);
s1 := Sym(144)!(  1, 77)(  2, 79)(  3, 78)(  4, 80)(  5, 73)(  6, 75)(  7, 74)(  8, 76)(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13,101)( 14,103)( 15,102)( 16,104)( 17, 97)( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)( 33, 93)( 34, 95)( 35, 94)( 36, 96)( 37,113)( 38,115)( 39,114)( 40,116)( 41,109)( 42,111)( 43,110)( 44,112)( 45,117)( 46,119)( 47,118)( 48,120)( 49,137)( 50,139)( 51,138)( 52,140)( 53,133)( 54,135)( 55,134)( 56,136)( 57,141)( 58,143)( 59,142)( 60,144)( 61,125)( 62,127)( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132);
s2 := Sym(144)!(  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 29)(  6, 32)(  7, 31)(  8, 30)(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 14, 16)( 18, 20)( 22, 24)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 65)( 42, 68)( 43, 67)( 44, 66)( 45, 69)( 46, 72)( 47, 71)( 48, 70)( 50, 52)( 54, 56)( 58, 60)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,101)( 78,104)( 79,103)( 80,102)( 81,105)( 82,108)( 83,107)( 84,106)( 86, 88)( 90, 92)( 94, 96)(109,133)(110,136)(111,135)(112,134)(113,137)(114,140)(115,139)(116,138)(117,141)(118,144)(119,143)(120,142)(122,124)(126,128)(130,132);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*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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