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}*768h
if this polytope has a name.
Group : SmallGroup(768,1088009)
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}*384e
   4-fold quotients : {3,6}*192, {12,6}*192a
   8-fold quotients : {6,6}*96
   16-fold quotients : {3,6}*48, {6,3}*48
   32-fold quotients : {3,3}*24
   48-fold quotients : {4,2}*16
   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*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 2.
      16 facets:
         16 of {12}*24
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      16 facets:
         16 of {12}*24
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s0*s2*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*s2*s1*s0*s1*s2*s1*s2*s1, s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*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> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      16 vertex figures:
         16 of {6}*12

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