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}*768k
if this polytope has a name.
Group : SmallGroup(768,1087795)
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}*384e
   4-fold quotients : {3,8}*192, {6,4}*192b
   8-fold quotients : {6,4}*96
   16-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   32-fold quotients : {3,4}*24, {6,2}*24
   48-fold quotients : {2,4}*16
   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*s2*s1*s2*s1*s2*s1*s0*s2*s1> 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*s2*s1*s2*s1*s0*s2*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*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      32 facets:
         32 of {6}*12
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      48 facets:
         32 of {3}*6
         16 of {6}*12
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s0*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*s0*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 4.
      24 facets:
         16 of {3}*6
         8 of {6}*12
      16 vertex figures:
         8 of {8}*16
         8 of {4}*8
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1> 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*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*s2*s1*s2*s1*s2*s1*s0*s2*s1, s1*s0*s1*s2*s1*s0*s1*s0*s2*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*s0*s2*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=<s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         8 of {8}*16
         16 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*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=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*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*s0*s2*s1*s2*s1, s0*s2*s1*s2*s1*s0*s2*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=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      12 vertex figures:
         4 of {8}*16
         8 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      8 vertex figures:
         4 of {8}*16
         4 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      8 vertex figures:
         4 of {8}*16
         4 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s2*s1> of order 8.
      12 facets:
         8 of {3}*6
         4 of {6}*12
      12 vertex figures:
         4 of {8}*16
         8 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         8 of {3}*6
         4 of {6}*12
      8 vertex figures:
         4 of {8}*16
         4 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*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=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      8 vertex figures:
         4 of {8}*16
         4 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         8 of {3}*6
         4 of {6}*12
      8 vertex figures:
         4 of {8}*16
         4 of {4}*8

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