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}*768k
if this polytope has a name.
Group : SmallGroup(768,1087795)
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}*384e
   4-fold quotients : {8,3}*192, {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=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*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*s2*s1*s0*s1*s2*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      48 vertex figures:
         32 of {3}*6
         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*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=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      24 vertex figures:
         16 of {3}*6
         8 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> 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*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> 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*s0*s1*s2*s1*s0*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=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s2*s1*s0*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*s2*s1*s0*s2*s1*s0*s1*s0*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*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s0*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, s1*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         8 of {2}*4
         4 of {8}*16
      12 vertex figures:
         8 of {3}*6
         4 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 8.
      8 facets:
         4 of {4}*8
         4 of {8}*16
      12 vertex figures:
         8 of {3}*6
         4 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
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s1*s2*s1*s0*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=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
      8 facets:
         4 of {4}*8
         4 of {8}*16
      12 vertex figures:
         8 of {3}*6
         4 of {6}*12

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

Twisty Puzzle