Polytope of Type {8,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*768q
if this polytope has a name.
Group : SmallGroup(768,1087633)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 32, 192, 48
Order of s0s1s2 : 12
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
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, {4,12}*192b
   8-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
   16-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {4,3}*24, {2,6}*24
   48-fold quotients : {2,4}*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*s2*s1*s0*s1*s0*s2> of order 2.
      24 facets:
         24 of {8}*16
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 2.
      24 facets:
         24 of {8}*16
      16 vertex figures:
         16 of {12}*24
   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
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      8 vertex figures:
         8 of {12}*24
   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
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s1> of order 4.
      24 facets:
         16 of {2}*4
         8 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      16 facets:
         8 of {4}*8
         8 of {8}*16
      8 vertex figures:
         8 of {12}*24

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

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