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}*768u
if this polytope has a name.
Group : SmallGroup(768,1087755)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 32, 192, 48
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
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 : {4,12}*384d, {8,6}*384g
   4-fold quotients : {8,12}*192a, {4,12}*192b, {4,6}*192b, {4,12}*192c
   8-fold quotients : {4,12}*96a, {8,6}*96, {4,12}*96b, {4,12}*96c, {4,6}*96
   12-fold quotients : {8,4}*64a
   16-fold quotients : {2,12}*48, {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   24-fold quotients : {4,4}*32, {8,2}*32
   32-fold quotients : {4,3}*24, {2,6}*24
   48-fold quotients : {2,4}*16, {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*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*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*s2*s1*s2*s1*s0*s2*s1*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*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*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*s2*s1*s0*s1*s2> of order 2.
      24 facets:
         24 of {8}*16
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      16 facets:
         16 of {8}*16
      16 vertex figures:
         8 of {4}*8
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 4.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24

Permutation Representation (GAP) : 
s0 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 73, 88)( 74, 87)( 75, 86)( 76, 85)( 77, 92)( 78, 91)( 79, 90)( 80, 89)( 81, 96)( 82, 95)( 83, 94)( 84, 93)( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)(104,149)(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)(112,157)(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)(120,165)(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)(128,173)(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)(136,181)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189);;
s1 := (  1,121)(  2,122)(  3,124)(  4,123)(  5,129)(  6,130)(  7,132)(  8,131)(  9,125)( 10,126)( 11,128)( 12,127)( 13,133)( 14,134)( 15,136)( 16,135)( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)( 24,139)( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)( 32,107)( 33,101)( 34,102)( 35,104)( 36,103)( 37,109)( 38,110)( 39,112)( 40,111)( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)( 96,151);;
s2 := (  1,  9)(  2, 11)(  3, 10)(  4, 12)(  6,  7)( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)( 52, 60)( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)( 90, 91)( 97,129)( 98,131)( 99,130)(100,132)(101,125)(102,127)(103,126)(104,128)(105,121)(106,123)(107,122)(108,124)(109,141)(110,143)(111,142)(112,144)(113,137)(114,139)(115,138)(116,140)(117,133)(118,135)(119,134)(120,136)(145,177)(146,179)(147,178)(148,180)(149,173)(150,175)(151,174)(152,176)(153,169)(154,171)(155,170)(156,172)(157,189)(158,191)(159,190)(160,192)(161,185)(162,187)(163,186)(164,188)(165,181)(166,183)(167,182)(168,184);;
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*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, 
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,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 73, 88)( 74, 87)( 75, 86)( 76, 85)( 77, 92)( 78, 91)( 79, 90)( 80, 89)( 81, 96)( 82, 95)( 83, 94)( 84, 93)( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)(104,149)(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)(112,157)(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)(120,165)(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)(128,173)(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)(136,181)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189);
s1 := Sym(192)!(  1,121)(  2,122)(  3,124)(  4,123)(  5,129)(  6,130)(  7,132)(  8,131)(  9,125)( 10,126)( 11,128)( 12,127)( 13,133)( 14,134)( 15,136)( 16,135)( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)( 24,139)( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)( 32,107)( 33,101)( 34,102)( 35,104)( 36,103)( 37,109)( 38,110)( 39,112)( 40,111)( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)( 96,151);
s2 := Sym(192)!(  1,  9)(  2, 11)(  3, 10)(  4, 12)(  6,  7)( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)( 52, 60)( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)( 90, 91)( 97,129)( 98,131)( 99,130)(100,132)(101,125)(102,127)(103,126)(104,128)(105,121)(106,123)(107,122)(108,124)(109,141)(110,143)(111,142)(112,144)(113,137)(114,139)(115,138)(116,140)(117,133)(118,135)(119,134)(120,136)(145,177)(146,179)(147,178)(148,180)(149,173)(150,175)(151,174)(152,176)(153,169)(154,171)(155,170)(156,172)(157,189)(158,191)(159,190)(160,192)(161,185)(162,187)(163,186)(164,188)(165,181)(166,183)(167,182)(168,184);
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*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, 
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

Twisty Puzzle