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}*768i
if this polytope has a name.
Group : SmallGroup(768,1088551)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 64, 192, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,6}*384c
   4-fold quotients : {6,6}*192a
   8-fold quotients : {6,6}*96
   16-fold quotients : {3,6}*48, {6,3}*48
   32-fold quotients : {3,3}*24
   96-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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      16 facets:
         16 of {12}*24
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 2.
      16 facets:
         16 of {12}*24
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      16 facets:
         16 of {12}*24
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12

Permutation Representation (GAP) : 
s0 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)( 99,100)(101,104)(102,103)(105,106)(107,108)(109,112)(110,111)(113,122)(114,121)(115,124)(116,123)(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)(132,163)(133,168)(134,167)(135,166)(136,165)(137,170)(138,169)(139,172)(140,171)(141,176)(142,175)(143,174)(144,173)(145,186)(146,185)(147,188)(148,187)(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)(156,179)(157,184)(158,183)(159,182)(160,181);;
s1 := (  1,161)(  2,162)(  3,165)(  4,166)(  5,163)(  6,164)(  7,167)(  8,168)(  9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121);;
s2 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 16)(  6, 15)(  7, 14)(  8, 13)( 21, 24)( 22, 23)( 29, 32)( 30, 31)( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)( 41, 65)( 42, 66)( 43, 67)( 44, 68)( 45, 72)( 46, 71)( 47, 70)( 48, 69)( 49, 81)( 50, 82)( 51, 83)( 52, 84)( 53, 88)( 54, 87)( 55, 86)( 56, 85)( 57, 89)( 58, 90)( 59, 91)( 60, 92)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 97,105)( 98,106)( 99,107)(100,108)(101,112)(102,111)(103,110)(104,109)(117,120)(118,119)(125,128)(126,127)(129,169)(130,170)(131,171)(132,172)(133,176)(134,175)(135,174)(136,173)(137,161)(138,162)(139,163)(140,164)(141,168)(142,167)(143,166)(144,165)(145,177)(146,178)(147,179)(148,180)(149,184)(150,183)(151,182)(152,181)(153,185)(154,186)(155,187)(156,188)(157,192)(158,191)(159,190)(160,189);;
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*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(192)!(  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)( 99,100)(101,104)(102,103)(105,106)(107,108)(109,112)(110,111)(113,122)(114,121)(115,124)(116,123)(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)(132,163)(133,168)(134,167)(135,166)(136,165)(137,170)(138,169)(139,172)(140,171)(141,176)(142,175)(143,174)(144,173)(145,186)(146,185)(147,188)(148,187)(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)(156,179)(157,184)(158,183)(159,182)(160,181);
s1 := Sym(192)!(  1,161)(  2,162)(  3,165)(  4,166)(  5,163)(  6,164)(  7,167)(  8,168)(  9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121);
s2 := Sym(192)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 16)(  6, 15)(  7, 14)(  8, 13)( 21, 24)( 22, 23)( 29, 32)( 30, 31)( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)( 41, 65)( 42, 66)( 43, 67)( 44, 68)( 45, 72)( 46, 71)( 47, 70)( 48, 69)( 49, 81)( 50, 82)( 51, 83)( 52, 84)( 53, 88)( 54, 87)( 55, 86)( 56, 85)( 57, 89)( 58, 90)( 59, 91)( 60, 92)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 97,105)( 98,106)( 99,107)(100,108)(101,112)(102,111)(103,110)(104,109)(117,120)(118,119)(125,128)(126,127)(129,169)(130,170)(131,171)(132,172)(133,176)(134,175)(135,174)(136,173)(137,161)(138,162)(139,163)(140,164)(141,168)(142,167)(143,166)(144,165)(145,177)(146,178)(147,179)(148,180)(149,184)(150,183)(151,182)(152,181)(153,185)(154,186)(155,187)(156,188)(157,192)(158,191)(159,190)(160,189);
poly := sub<Sym(192)|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*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 >;
References : None.
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