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}*1944b
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 243, 486, 81
Order of s0s1s2 : 6
Order of s0s1s2s1 : 9
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) :
   9-fold quotients : {12,4}*216
   27-fold quotients : {4,4}*72
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> of order 3.
      27 facets:
         27 of {12}*24
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {12}*24
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      33 facets:
         24 of {12}*24
         9 of {4}*8
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      27 facets:
         27 of {12}*24
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 9.
      15 facets:
         6 of {12}*24
         9 of {4}*8
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 9.
      15 facets:
         6 of {12}*24
         9 of {4}*8
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 9.
      9 facets:
         9 of {12}*24
      27 vertex figures:
         27 of {4}*8

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,184)( 29,186)( 30,185)( 31,189)( 32,188)( 33,187)( 34,182)( 35,181)( 36,183)( 37,179)( 38,178)( 39,180)( 40,172)( 41,174)( 42,173)( 43,177)( 44,176)( 45,175)( 46,165)( 47,164)( 48,163)( 49,167)( 50,166)( 51,168)( 52,169)( 53,171)( 54,170)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)( 64, 86)( 65, 85)( 66, 87)( 67, 88)( 68, 90)( 69, 89)( 70, 84)( 71, 83)( 72, 82)( 73,108)( 74,107)( 75,106)( 76,101)( 77,100)( 78,102)( 79,103)( 80,105)( 81,104)(109,226)(110,228)(111,227)(112,231)(113,230)(114,229)(115,233)(116,232)(117,234)(118,221)(119,220)(120,222)(121,223)(122,225)(123,224)(124,219)(125,218)(126,217)(127,243)(128,242)(129,241)(130,236)(131,235)(132,237)(133,238)(134,240)(135,239)(136,149)(137,148)(138,150)(139,151)(140,153)(141,152)(142,147)(143,146)(144,145)(155,156)(157,159)(160,161)(190,192)(193,194)(197,198)(199,211)(200,213)(201,212)(202,216)(203,215)(204,214)(205,209)(206,208)(207,210);;
s1 := (  1, 10)(  2, 18)(  3, 14)(  4, 13)(  5, 12)(  6, 17)(  7, 16)(  8, 15)(  9, 11)( 20, 27)( 21, 23)( 24, 26)( 28, 53)( 29, 49)( 30, 48)( 31, 47)( 32, 52)( 33, 51)( 34, 50)( 35, 46)( 36, 54)( 37, 44)( 38, 40)( 41, 43)( 55, 62)( 56, 58)( 59, 61)( 64, 80)( 65, 76)( 66, 75)( 67, 74)( 68, 79)( 69, 78)( 70, 77)( 71, 73)( 72, 81)( 82,172)( 83,180)( 84,176)( 85,175)( 86,174)( 87,179)( 88,178)( 89,177)( 90,173)( 91,163)( 92,171)( 93,167)( 94,166)( 95,165)( 96,170)( 97,169)( 98,168)( 99,164)(100,181)(101,189)(102,185)(103,184)(104,183)(105,188)(106,187)(107,186)(108,182)(109,215)(110,211)(111,210)(112,209)(113,214)(114,213)(115,212)(116,208)(117,216)(118,206)(119,202)(120,201)(121,200)(122,205)(123,204)(124,203)(125,199)(126,207)(127,197)(128,193)(129,192)(130,191)(131,196)(132,195)(133,194)(134,190)(135,198)(136,224)(137,220)(138,219)(139,218)(140,223)(141,222)(142,221)(143,217)(144,225)(145,242)(146,238)(147,237)(148,236)(149,241)(150,240)(151,239)(152,235)(153,243)(154,233)(155,229)(156,228)(157,227)(158,232)(159,231)(160,230)(161,226)(162,234);;
s2 := (  1,191)(  2,192)(  3,190)(  4,198)(  5,196)(  6,197)(  7,193)(  8,194)(  9,195)( 10,208)( 11,209)( 12,210)( 13,215)( 14,216)( 15,214)( 16,213)( 17,211)( 18,212)( 19,201)( 20,199)( 21,200)( 22,205)( 23,206)( 24,207)( 25,203)( 26,204)( 27,202)( 28, 47)( 29, 48)( 30, 46)( 31, 54)( 32, 52)( 33, 53)( 34, 49)( 35, 50)( 36, 51)( 40, 44)( 41, 45)( 42, 43)( 55,125)( 56,126)( 57,124)( 58,123)( 59,121)( 60,122)( 61,118)( 62,119)( 63,120)( 64,115)( 65,116)( 66,117)( 67,113)( 68,114)( 69,112)( 70,111)( 71,109)( 72,110)( 73,135)( 74,133)( 75,134)( 76,130)( 77,131)( 78,132)( 79,128)( 80,129)( 81,127)( 82,228)( 83,226)( 84,227)( 85,232)( 86,233)( 87,234)( 88,230)( 89,231)( 90,229)( 91,218)( 92,219)( 93,217)( 94,225)( 95,223)( 96,224)( 97,220)( 98,221)( 99,222)(100,235)(101,236)(102,237)(103,242)(104,243)(105,241)(106,240)(107,238)(108,239)(136,148)(137,149)(138,150)(139,146)(140,147)(141,145)(142,153)(143,151)(144,152)(154,158)(155,159)(156,157)(163,186)(164,184)(165,185)(166,181)(167,182)(168,183)(169,188)(170,189)(171,187)(172,176)(173,177)(174,175);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(243)!(  2,  3)(  4,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,184)( 29,186)( 30,185)( 31,189)( 32,188)( 33,187)( 34,182)( 35,181)( 36,183)( 37,179)( 38,178)( 39,180)( 40,172)( 41,174)( 42,173)( 43,177)( 44,176)( 45,175)( 46,165)( 47,164)( 48,163)( 49,167)( 50,166)( 51,168)( 52,169)( 53,171)( 54,170)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)( 64, 86)( 65, 85)( 66, 87)( 67, 88)( 68, 90)( 69, 89)( 70, 84)( 71, 83)( 72, 82)( 73,108)( 74,107)( 75,106)( 76,101)( 77,100)( 78,102)( 79,103)( 80,105)( 81,104)(109,226)(110,228)(111,227)(112,231)(113,230)(114,229)(115,233)(116,232)(117,234)(118,221)(119,220)(120,222)(121,223)(122,225)(123,224)(124,219)(125,218)(126,217)(127,243)(128,242)(129,241)(130,236)(131,235)(132,237)(133,238)(134,240)(135,239)(136,149)(137,148)(138,150)(139,151)(140,153)(141,152)(142,147)(143,146)(144,145)(155,156)(157,159)(160,161)(190,192)(193,194)(197,198)(199,211)(200,213)(201,212)(202,216)(203,215)(204,214)(205,209)(206,208)(207,210);
s1 := Sym(243)!(  1, 10)(  2, 18)(  3, 14)(  4, 13)(  5, 12)(  6, 17)(  7, 16)(  8, 15)(  9, 11)( 20, 27)( 21, 23)( 24, 26)( 28, 53)( 29, 49)( 30, 48)( 31, 47)( 32, 52)( 33, 51)( 34, 50)( 35, 46)( 36, 54)( 37, 44)( 38, 40)( 41, 43)( 55, 62)( 56, 58)( 59, 61)( 64, 80)( 65, 76)( 66, 75)( 67, 74)( 68, 79)( 69, 78)( 70, 77)( 71, 73)( 72, 81)( 82,172)( 83,180)( 84,176)( 85,175)( 86,174)( 87,179)( 88,178)( 89,177)( 90,173)( 91,163)( 92,171)( 93,167)( 94,166)( 95,165)( 96,170)( 97,169)( 98,168)( 99,164)(100,181)(101,189)(102,185)(103,184)(104,183)(105,188)(106,187)(107,186)(108,182)(109,215)(110,211)(111,210)(112,209)(113,214)(114,213)(115,212)(116,208)(117,216)(118,206)(119,202)(120,201)(121,200)(122,205)(123,204)(124,203)(125,199)(126,207)(127,197)(128,193)(129,192)(130,191)(131,196)(132,195)(133,194)(134,190)(135,198)(136,224)(137,220)(138,219)(139,218)(140,223)(141,222)(142,221)(143,217)(144,225)(145,242)(146,238)(147,237)(148,236)(149,241)(150,240)(151,239)(152,235)(153,243)(154,233)(155,229)(156,228)(157,227)(158,232)(159,231)(160,230)(161,226)(162,234);
s2 := Sym(243)!(  1,191)(  2,192)(  3,190)(  4,198)(  5,196)(  6,197)(  7,193)(  8,194)(  9,195)( 10,208)( 11,209)( 12,210)( 13,215)( 14,216)( 15,214)( 16,213)( 17,211)( 18,212)( 19,201)( 20,199)( 21,200)( 22,205)( 23,206)( 24,207)( 25,203)( 26,204)( 27,202)( 28, 47)( 29, 48)( 30, 46)( 31, 54)( 32, 52)( 33, 53)( 34, 49)( 35, 50)( 36, 51)( 40, 44)( 41, 45)( 42, 43)( 55,125)( 56,126)( 57,124)( 58,123)( 59,121)( 60,122)( 61,118)( 62,119)( 63,120)( 64,115)( 65,116)( 66,117)( 67,113)( 68,114)( 69,112)( 70,111)( 71,109)( 72,110)( 73,135)( 74,133)( 75,134)( 76,130)( 77,131)( 78,132)( 79,128)( 80,129)( 81,127)( 82,228)( 83,226)( 84,227)( 85,232)( 86,233)( 87,234)( 88,230)( 89,231)( 90,229)( 91,218)( 92,219)( 93,217)( 94,225)( 95,223)( 96,224)( 97,220)( 98,221)( 99,222)(100,235)(101,236)(102,237)(103,242)(104,243)(105,241)(106,240)(107,238)(108,239)(136,148)(137,149)(138,150)(139,146)(140,147)(141,145)(142,153)(143,151)(144,152)(154,158)(155,159)(156,157)(163,186)(164,184)(165,185)(166,181)(167,182)(168,183)(169,188)(170,189)(171,187)(172,176)(173,177)(174,175);
poly := sub<Sym(243)|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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope

Twisty Puzzle