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}*768e
if this polytope has a name.
Group : SmallGroup(768,1086320)
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 : {4,6}*384a
   4-fold quotients : {4,6}*192a, {8,6}*192b
   8-fold quotients : {8,3}*96, {4,6}*96
   16-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {4,3}*24, {2,6}*24
   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*s2*s1*s0*s1*s2*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=<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*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*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
      40 vertex figures:
         16 of {3}*6
         24 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 4.
      12 facets:
         12 of {8}*16
      24 vertex figures:
         16 of {3}*6
         8 of {6}*12
   P/N, where N=<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=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      12 facets:
         12 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*s2*s1*s0*s2*s1*s0*s1*s2*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*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
      6 facets:
         6 of {8}*16
      12 vertex figures:
         8 of {3}*6
         4 of {6}*12

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

Twisty Puzzle