Polytope of Type {18,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1944c
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 81, 486, 54
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
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 : {6,12}*216b
   27-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 3.
      36 facets:
         9 of {18}*36
         27 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      27 vertex figures:
         27 of {12}*24

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