Polytope of Type {12,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1944h
if this polytope has a name.
Group : SmallGroup(1944,2325)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 54, 486, 81
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {12,6}*216c
   27-fold quotients : {4,6}*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*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      27 facets:
         27 of {12}*24
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      27 facets:
         27 of {12}*24
      36 vertex figures:
         9 of {18}*36
         27 of {6}*12

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