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}*864d
if this polytope has a name.
Group : SmallGroup(864,4080)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 108, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {12,4,2} of size 1728
Vertex Figure Of :
   {2,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*432b
   3-fold quotients : {12,4}*288
   6-fold quotients : {6,4}*144
   9-fold quotients : {12,4}*96a
   12-fold quotients : {6,4}*72
   18-fold quotients : {12,2}*48, {6,4}*48a
   27-fold quotients : {4,4}*32
   36-fold quotients : {6,2}*24
   54-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,4}*1728f, {24,4}*1728g, {12,8}*1728g, {12,8}*1728h, {12,4}*1728c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {12}*24
      60 vertex figures:
         48 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      18 facets:
         9 of {4}*8
         9 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 6.
      6 facets:
         6 of {12}*24
      24 vertex figures:
         12 of {4}*8
         12 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 6.
      6 facets:
         6 of {12}*24
      24 vertex figures:
         12 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      6 facets:
         3 of {12}*24
         3 of {4}*8
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s1*s2> of order 9.
      8 facets:
         6 of {4}*8
         2 of {12}*24
      12 vertex figures:
         12 of {4}*8

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

Twisty Puzzle