Polytope of Type {12,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,12630)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 216, 432, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*864b
   3-fold quotients : {12,4}*576
   4-fold quotients : {6,4}*432a
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,4}*216
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {4,4}*64
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   216-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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 2.
      36 facets:
         36 of {12}*24
      114 vertex figures:
         102 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 2.
      54 facets:
         18 of {12}*24
         36 of {6}*12
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      32 facets:
         20 of {12}*24
         12 of {4}*8
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2, s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
      28 facets:
         9 of {12}*24
         17 of {6}*12
         2 of {3}*6
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      26 facets:
         8 of {6}*12
         6 of {12}*24
         12 of {2}*4
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1> of order 6.
      12 facets:
         12 of {12}*24
      42 vertex figures:
         30 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      22 facets:
         4 of {12}*24
         12 of {6}*12
         6 of {4}*8
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 6.
      18 facets:
         6 of {12}*24
         12 of {6}*12
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1, s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 6.
      16 facets:
         10 of {12}*24
         6 of {4}*8
      42 vertex figures:
         30 of {4}*8
         12 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 12.
      12 facets:
         2 of {12}*24
         5 of {6}*12
         2 of {3}*6
         3 of {4}*8
      18 vertex figures:
         18 of {4}*8

Permutation Representation (GAP) : 
s0 := (  1,136)(  2,137)(  3,138)(  4,144)(  5,142)(  6,143)(  7,140)(  8,141)(  9,139)( 10,156)( 11,154)( 12,155)( 13,161)( 14,162)( 15,160)( 16,157)( 17,158)( 18,159)( 19,146)( 20,147)( 21,145)( 22,151)( 23,152)( 24,153)( 25,150)( 26,148)( 27,149)( 28,109)( 29,110)( 30,111)( 31,117)( 32,115)( 33,116)( 34,113)( 35,114)( 36,112)( 37,129)( 38,127)( 39,128)( 40,134)( 41,135)( 42,133)( 43,130)( 44,131)( 45,132)( 46,119)( 47,120)( 48,118)( 49,124)( 50,125)( 51,126)( 52,123)( 53,121)( 54,122)( 55,190)( 56,191)( 57,192)( 58,198)( 59,196)( 60,197)( 61,194)( 62,195)( 63,193)( 64,210)( 65,208)( 66,209)( 67,215)( 68,216)( 69,214)( 70,211)( 71,212)( 72,213)( 73,200)( 74,201)( 75,199)( 76,205)( 77,206)( 78,207)( 79,204)( 80,202)( 81,203)( 82,163)( 83,164)( 84,165)( 85,171)( 86,169)( 87,170)( 88,167)( 89,168)( 90,166)( 91,183)( 92,181)( 93,182)( 94,188)( 95,189)( 96,187)( 97,184)( 98,185)( 99,186)(100,173)(101,174)(102,172)(103,178)(104,179)(105,180)(106,177)(107,175)(108,176);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,102)(103,108)(104,107)(105,106)(109,193)(110,195)(111,194)(112,190)(113,192)(114,191)(115,196)(116,198)(117,197)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,210)(128,209)(129,208)(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)(136,166)(137,168)(138,167)(139,163)(140,165)(141,164)(142,169)(143,171)(144,170)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(154,183)(155,182)(156,181)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184);;
s2 := (  1,136)(  2,138)(  3,137)(  4,151)(  5,153)(  6,152)(  7,159)(  8,158)(  9,157)( 10,146)( 11,145)( 12,147)( 13,161)( 14,160)( 15,162)( 16,139)( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,144)( 23,143)( 24,142)( 25,149)( 26,148)( 27,150)( 28,109)( 29,111)( 30,110)( 31,124)( 32,126)( 33,125)( 34,132)( 35,131)( 36,130)( 37,119)( 38,118)( 39,120)( 40,134)( 41,133)( 42,135)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)( 49,117)( 50,116)( 51,115)( 52,122)( 53,121)( 54,123)( 55,163)( 56,165)( 57,164)( 58,178)( 59,180)( 60,179)( 61,186)( 62,185)( 63,184)( 64,173)( 65,172)( 66,174)( 67,188)( 68,187)( 69,189)( 70,166)( 71,168)( 72,167)( 73,183)( 74,182)( 75,181)( 76,171)( 77,170)( 78,169)( 79,176)( 80,175)( 81,177)( 82,190)( 83,192)( 84,191)( 85,205)( 86,207)( 87,206)( 88,213)( 89,212)( 90,211)( 91,200)( 92,199)( 93,201)( 94,215)( 95,214)( 96,216)( 97,193)( 98,195)( 99,194)(100,210)(101,209)(102,208)(103,198)(104,197)(105,196)(106,203)(107,202)(108,204);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(216)!(  1,136)(  2,137)(  3,138)(  4,144)(  5,142)(  6,143)(  7,140)(  8,141)(  9,139)( 10,156)( 11,154)( 12,155)( 13,161)( 14,162)( 15,160)( 16,157)( 17,158)( 18,159)( 19,146)( 20,147)( 21,145)( 22,151)( 23,152)( 24,153)( 25,150)( 26,148)( 27,149)( 28,109)( 29,110)( 30,111)( 31,117)( 32,115)( 33,116)( 34,113)( 35,114)( 36,112)( 37,129)( 38,127)( 39,128)( 40,134)( 41,135)( 42,133)( 43,130)( 44,131)( 45,132)( 46,119)( 47,120)( 48,118)( 49,124)( 50,125)( 51,126)( 52,123)( 53,121)( 54,122)( 55,190)( 56,191)( 57,192)( 58,198)( 59,196)( 60,197)( 61,194)( 62,195)( 63,193)( 64,210)( 65,208)( 66,209)( 67,215)( 68,216)( 69,214)( 70,211)( 71,212)( 72,213)( 73,200)( 74,201)( 75,199)( 76,205)( 77,206)( 78,207)( 79,204)( 80,202)( 81,203)( 82,163)( 83,164)( 84,165)( 85,171)( 86,169)( 87,170)( 88,167)( 89,168)( 90,166)( 91,183)( 92,181)( 93,182)( 94,188)( 95,189)( 96,187)( 97,184)( 98,185)( 99,186)(100,173)(101,174)(102,172)(103,178)(104,179)(105,180)(106,177)(107,175)(108,176);
s1 := Sym(216)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,102)(103,108)(104,107)(105,106)(109,193)(110,195)(111,194)(112,190)(113,192)(114,191)(115,196)(116,198)(117,197)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,210)(128,209)(129,208)(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)(136,166)(137,168)(138,167)(139,163)(140,165)(141,164)(142,169)(143,171)(144,170)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(154,183)(155,182)(156,181)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184);
s2 := Sym(216)!(  1,136)(  2,138)(  3,137)(  4,151)(  5,153)(  6,152)(  7,159)(  8,158)(  9,157)( 10,146)( 11,145)( 12,147)( 13,161)( 14,160)( 15,162)( 16,139)( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,144)( 23,143)( 24,142)( 25,149)( 26,148)( 27,150)( 28,109)( 29,111)( 30,110)( 31,124)( 32,126)( 33,125)( 34,132)( 35,131)( 36,130)( 37,119)( 38,118)( 39,120)( 40,134)( 41,133)( 42,135)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)( 49,117)( 50,116)( 51,115)( 52,122)( 53,121)( 54,123)( 55,163)( 56,165)( 57,164)( 58,178)( 59,180)( 60,179)( 61,186)( 62,185)( 63,184)( 64,173)( 65,172)( 66,174)( 67,188)( 68,187)( 69,189)( 70,166)( 71,168)( 72,167)( 73,183)( 74,182)( 75,181)( 76,171)( 77,170)( 78,169)( 79,176)( 80,175)( 81,177)( 82,190)( 83,192)( 84,191)( 85,205)( 86,207)( 87,206)( 88,213)( 89,212)( 90,211)( 91,200)( 92,199)( 93,201)( 94,215)( 95,214)( 96,216)( 97,193)( 98,195)( 99,194)(100,210)(101,209)(102,208)(103,198)(104,197)(105,196)(106,203)(107,202)(108,204);
poly := sub<Sym(216)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle