Polytope of Type {6,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8}*768f
if this polytope has a name.
Group : SmallGroup(768,1086324)
Rank : 3
Schlafli Type : {6,8}
Number of vertices, edges, etc : 48, 192, 64
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 : {6,8}*384d
   4-fold quotients : {6,8}*192a, {6,8}*192b
   8-fold quotients : {3,8}*96, {6,4}*96
   16-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   32-fold quotients : {3,4}*24, {6,2}*24
   64-fold quotients : {3,2}*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=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
      32 facets:
         32 of {6}*12
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 2.
      32 facets:
         32 of {6}*12
      32 vertex figures:
         16 of {8}*16
         16 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      40 facets:
         16 of {3}*6
         24 of {6}*12
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 2.
      32 facets:
         32 of {6}*12
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
      16 facets:
         16 of {6}*12
      16 vertex figures:
         8 of {8}*16
         8 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 4.
      16 facets:
         16 of {6}*12
      16 vertex figures:
         8 of {8}*16
         8 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
      16 facets:
         16 of {6}*12
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 4.
      16 facets:
         16 of {6}*12
      20 vertex figures:
         8 of {8}*16
         8 of {2}*4
         4 of {4}*8
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 4.
      16 facets:
         16 of {6}*12
      20 vertex figures:
         16 of {4}*8
         4 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 4.
      24 facets:
         16 of {3}*6
         8 of {6}*12
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      10 vertex figures:
         4 of {8}*16
         4 of {2}*4
         2 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 8.
      8 facets:
         8 of {6}*12
      8 vertex figures:
         4 of {8}*16
         4 of {4}*8
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      10 vertex figures:
         8 of {4}*8
         2 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      12 vertex figures:
         6 of {4}*8
         2 of {8}*16
         4 of {2}*4

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

Twisty Puzzle