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}*576d
if this polytope has a name.
Group : SmallGroup(576,8340)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 48, 144, 24
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
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 : {6,6}*288b
   3-fold quotients : {12,6}*192b
   4-fold quotients : {3,6}*144
   6-fold quotients : {12,3}*96, {6,6}*96
   8-fold quotients : {6,6}*72c
   12-fold quotients : {3,6}*48, {6,3}*48
   16-fold quotients : {3,6}*36
   24-fold quotients : {3,3}*24, {6,2}*24
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12}*1152e, {12,6}*1152a, {12,12}*1152q
   3-fold covers : {36,6}*1728c, {12,6}*1728d, {12,6}*1728g
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 2.
      12 facets:
         12 of {12}*24
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      12 facets:
         12 of {12}*24
      32 vertex figures:
         16 of {6}*12
         16 of {3}*6
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 3.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         12 of {6}*12
         12 of {2}*4

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