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

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

Twisty Puzzle