Polytope of Type {4,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*768d
if this polytope has a name.
Group : SmallGroup(768,1087743)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 32, 192, 96
Order of s0s1s2 : 12
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*384d
   4-fold quotients : {4,12}*192a, {4,12}*192b, {4,6}*192b, {4,12}*192c
   8-fold quotients : {4,12}*96a, {4,12}*96b, {4,12}*96c, {4,6}*96
   12-fold quotients : {4,4}*64
   16-fold quotients : {2,12}*48, {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   24-fold quotients : {4,4}*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=<s2*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 2.
      56 facets:
         40 of {4}*8
         16 of {2}*4
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      48 facets:
         48 of {4}*8
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      48 facets:
         48 of {4}*8
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      16 vertex figures:
         16 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      24 vertex figures:
         16 of {6}*12
         8 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      32 facets:
         32 of {4}*8
      16 vertex figures:
         8 of {4}*8
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      32 facets:
         16 of {4}*8
         16 of {2}*4
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      28 facets:
         20 of {4}*8
         8 of {2}*4
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         8 of {6}*12
         4 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         8 of {6}*12
         4 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         8 of {6}*12
         4 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      28 facets:
         20 of {4}*8
         8 of {2}*4
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      32 facets:
         16 of {4}*8
         16 of {2}*4
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         8 of {3}*6
         4 of {12}*24
         4 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      28 facets:
         8 of {2}*4
         20 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s1*s2*s1*s2> of order 6.
      16 facets:
         16 of {4}*8
      12 vertex figures:
         4 of {2}*4
         2 of {4}*8
         2 of {12}*24
         4 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         4 of {6}*12
         2 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         4 of {6}*12
         2 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
      18 facets:
         6 of {4}*8
         12 of {2}*4
      4 vertex figures:
         4 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, 40)( 26, 39)( 27, 38)( 28, 37)( 29, 44)( 30, 43)( 31, 42)( 32, 41)( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 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)(121,184)(122,183)(123,182)(124,181)(125,188)(126,187)(127,186)(128,185)(129,192)(130,191)(131,190)(132,189)(133,172)(134,171)(135,170)(136,169)(137,176)(138,175)(139,174)(140,173)(141,180)(142,179)(143,178)(144,177);;
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)( 49,169)( 50,170)( 51,172)( 52,171)( 53,177)( 54,178)( 55,180)( 56,179)( 57,173)( 58,174)( 59,176)( 60,175)( 61,181)( 62,182)( 63,184)( 64,183)( 65,189)( 66,190)( 67,192)( 68,191)( 69,185)( 70,186)( 71,188)( 72,187)( 73,145)( 74,146)( 75,148)( 76,147)( 77,153)( 78,154)( 79,156)( 80,155)( 81,149)( 82,150)( 83,152)( 84,151)( 85,157)( 86,158)( 87,160)( 88,159)( 89,165)( 90,166)( 91,168)( 92,167)( 93,161)( 94,162)( 95,164)( 96,163);;
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, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)( 58, 63)( 59, 62)( 60, 64)( 73, 93)( 74, 95)( 75, 94)( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)( 81, 85)( 82, 87)( 83, 86)( 84, 88)( 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)(145,189)(146,191)(147,190)(148,192)(149,185)(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,177)(158,179)(159,178)(160,180)(161,173)(162,175)(163,174)(164,176)(165,169)(166,171)(167,170)(168,172);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*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,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 40)( 26, 39)( 27, 38)( 28, 37)( 29, 44)( 30, 43)( 31, 42)( 32, 41)( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 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)(121,184)(122,183)(123,182)(124,181)(125,188)(126,187)(127,186)(128,185)(129,192)(130,191)(131,190)(132,189)(133,172)(134,171)(135,170)(136,169)(137,176)(138,175)(139,174)(140,173)(141,180)(142,179)(143,178)(144,177);
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)( 49,169)( 50,170)( 51,172)( 52,171)( 53,177)( 54,178)( 55,180)( 56,179)( 57,173)( 58,174)( 59,176)( 60,175)( 61,181)( 62,182)( 63,184)( 64,183)( 65,189)( 66,190)( 67,192)( 68,191)( 69,185)( 70,186)( 71,188)( 72,187)( 73,145)( 74,146)( 75,148)( 76,147)( 77,153)( 78,154)( 79,156)( 80,155)( 81,149)( 82,150)( 83,152)( 84,151)( 85,157)( 86,158)( 87,160)( 88,159)( 89,165)( 90,166)( 91,168)( 92,167)( 93,161)( 94,162)( 95,164)( 96,163);
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, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)( 58, 63)( 59, 62)( 60, 64)( 73, 93)( 74, 95)( 75, 94)( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)( 81, 85)( 82, 87)( 83, 86)( 84, 88)( 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)(145,189)(146,191)(147,190)(148,192)(149,185)(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,177)(158,179)(159,178)(160,180)(161,173)(162,175)(163,174)(164,176)(165,169)(166,171)(167,170)(168,172);
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*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

Twisty Puzzle