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}*1944f
if this polytope has a name.
Group : SmallGroup(1944,806)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 54, 486, 81
Order of s0s1s2 : 4
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}*216a
   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*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {12}*24
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      27 facets:
         27 of {12}*24
      36 vertex figures:
         27 of {6}*12
         9 of {18}*36

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