Polytope of Type {8,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6}*768l
if this polytope has a name.
Group : SmallGroup(768,1088009)
Rank : 3
Schlafli Type : {8,6}
Number of vertices, edges, etc : 64, 192, 48
Order of s0s1s2 : 12
Order of s0s1s2s1 : 8
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 : {8,6}*384d
   4-fold quotients : {8,6}*192a, {4,6}*192b
   8-fold quotients : {4,6}*96
   16-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {4,3}*24, {2,6}*24
   48-fold quotients : {4,2}*16
   64-fold quotients : {2,3}*12
   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=<s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      32 facets:
         16 of {4}*8
         16 of {8}*16
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 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*s2*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      12 facets:
         12 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 4.
      24 facets:
         16 of {2}*4
         8 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 8.
      8 facets:
         4 of {4}*8
         4 of {8}*16
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1> of order 8.
      8 facets:
         4 of {4}*8
         4 of {8}*16
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
      12 facets:
         8 of {2}*4
         4 of {8}*16
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 8.
      8 facets:
         4 of {4}*8
         4 of {8}*16
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 8.
      8 facets:
         4 of {4}*8
         4 of {8}*16
      8 vertex figures:
         8 of {6}*12

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