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}*1944a
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 81, 486, 54
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,4}*648
   9-fold quotients : {6,12}*216a
   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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      36 facets:
         27 of {6}*12
         9 of {18}*36
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {18}*36
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 78)( 29, 77)( 30, 76)( 31, 75)( 32, 74)( 33, 73)( 34, 81)( 35, 80)( 36, 79)( 37, 60)( 38, 59)( 39, 58)( 40, 57)( 41, 56)( 42, 55)( 43, 63)( 44, 62)( 45, 61)( 46, 69)( 47, 68)( 48, 67)( 49, 66)( 50, 65)( 51, 64)( 52, 72)( 53, 71)( 54, 70)( 82,169)( 83,171)( 84,170)( 85,166)( 86,168)( 87,167)( 88,163)( 89,165)( 90,164)( 91,178)( 92,180)( 93,179)( 94,175)( 95,177)( 96,176)( 97,172)( 98,174)( 99,173)(100,187)(101,189)(102,188)(103,184)(104,186)(105,185)(106,181)(107,183)(108,182)(109,237)(110,236)(111,235)(112,243)(113,242)(114,241)(115,240)(116,239)(117,238)(118,219)(119,218)(120,217)(121,225)(122,224)(123,223)(124,222)(125,221)(126,220)(127,228)(128,227)(129,226)(130,234)(131,233)(132,232)(133,231)(134,230)(135,229)(136,201)(137,200)(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,210)(146,209)(147,208)(148,216)(149,215)(150,214)(151,213)(152,212)(153,211)(154,192)(155,191)(156,190)(157,198)(158,197)(159,196)(160,195)(161,194)(162,193);;
s1 := (  1, 82)(  2, 86)(  3, 90)(  4, 88)(  5, 83)(  6, 87)(  7, 85)(  8, 89)(  9, 84)( 10,100)( 11,104)( 12,108)( 13,106)( 14,101)( 15,105)( 16,103)( 17,107)( 18,102)( 19, 91)( 20, 95)( 21, 99)( 22, 97)( 23, 92)( 24, 96)( 25, 94)( 26, 98)( 27, 93)( 28,130)( 29,134)( 30,129)( 31,127)( 32,131)( 33,135)( 34,133)( 35,128)( 36,132)( 37,121)( 38,125)( 39,120)( 40,118)( 41,122)( 42,126)( 43,124)( 44,119)( 45,123)( 46,112)( 47,116)( 48,111)( 49,109)( 50,113)( 51,117)( 52,115)( 53,110)( 54,114)( 55,151)( 56,146)( 57,150)( 58,148)( 59,152)( 60,147)( 61,145)( 62,149)( 63,153)( 64,142)( 65,137)( 66,141)( 67,139)( 68,143)( 69,138)( 70,136)( 71,140)( 72,144)( 73,160)( 74,155)( 75,159)( 76,157)( 77,161)( 78,156)( 79,154)( 80,158)( 81,162)(163,169)(165,168)(167,170)(172,187)(173,182)(174,186)(175,184)(176,188)(177,183)(178,181)(179,185)(180,189)(190,208)(191,212)(192,216)(193,214)(194,209)(195,213)(196,211)(197,215)(198,210)(200,203)(201,207)(202,205)(217,229)(218,233)(219,228)(220,226)(221,230)(222,234)(223,232)(224,227)(225,231)(235,238)(236,242)(240,243);;
s2 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28,186)( 29,185)( 30,184)( 31,188)( 32,187)( 33,189)( 34,181)( 35,183)( 36,182)( 37,177)( 38,176)( 39,175)( 40,179)( 41,178)( 42,180)( 43,172)( 44,174)( 45,173)( 46,168)( 47,167)( 48,166)( 49,170)( 50,169)( 51,171)( 52,163)( 53,165)( 54,164)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)( 64, 83)( 65, 82)( 66, 84)( 67, 85)( 68, 87)( 69, 86)( 70, 90)( 71, 89)( 72, 88)( 73,101)( 74,100)( 75,102)( 76,103)( 77,105)( 78,104)( 79,108)( 80,107)( 81,106)(109,229)(110,231)(111,230)(112,234)(113,233)(114,232)(115,227)(116,226)(117,228)(118,220)(119,222)(120,221)(121,225)(122,224)(123,223)(124,218)(125,217)(126,219)(127,238)(128,240)(129,239)(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,147)(137,146)(138,145)(139,149)(140,148)(141,150)(142,151)(143,153)(144,152)(154,156)(157,158)(161,162)(190,192)(193,194)(197,198)(199,210)(200,209)(201,208)(202,212)(203,211)(204,213)(205,214)(206,216)(207,215);;
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*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(243)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 78)( 29, 77)( 30, 76)( 31, 75)( 32, 74)( 33, 73)( 34, 81)( 35, 80)( 36, 79)( 37, 60)( 38, 59)( 39, 58)( 40, 57)( 41, 56)( 42, 55)( 43, 63)( 44, 62)( 45, 61)( 46, 69)( 47, 68)( 48, 67)( 49, 66)( 50, 65)( 51, 64)( 52, 72)( 53, 71)( 54, 70)( 82,169)( 83,171)( 84,170)( 85,166)( 86,168)( 87,167)( 88,163)( 89,165)( 90,164)( 91,178)( 92,180)( 93,179)( 94,175)( 95,177)( 96,176)( 97,172)( 98,174)( 99,173)(100,187)(101,189)(102,188)(103,184)(104,186)(105,185)(106,181)(107,183)(108,182)(109,237)(110,236)(111,235)(112,243)(113,242)(114,241)(115,240)(116,239)(117,238)(118,219)(119,218)(120,217)(121,225)(122,224)(123,223)(124,222)(125,221)(126,220)(127,228)(128,227)(129,226)(130,234)(131,233)(132,232)(133,231)(134,230)(135,229)(136,201)(137,200)(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,210)(146,209)(147,208)(148,216)(149,215)(150,214)(151,213)(152,212)(153,211)(154,192)(155,191)(156,190)(157,198)(158,197)(159,196)(160,195)(161,194)(162,193);
s1 := Sym(243)!(  1, 82)(  2, 86)(  3, 90)(  4, 88)(  5, 83)(  6, 87)(  7, 85)(  8, 89)(  9, 84)( 10,100)( 11,104)( 12,108)( 13,106)( 14,101)( 15,105)( 16,103)( 17,107)( 18,102)( 19, 91)( 20, 95)( 21, 99)( 22, 97)( 23, 92)( 24, 96)( 25, 94)( 26, 98)( 27, 93)( 28,130)( 29,134)( 30,129)( 31,127)( 32,131)( 33,135)( 34,133)( 35,128)( 36,132)( 37,121)( 38,125)( 39,120)( 40,118)( 41,122)( 42,126)( 43,124)( 44,119)( 45,123)( 46,112)( 47,116)( 48,111)( 49,109)( 50,113)( 51,117)( 52,115)( 53,110)( 54,114)( 55,151)( 56,146)( 57,150)( 58,148)( 59,152)( 60,147)( 61,145)( 62,149)( 63,153)( 64,142)( 65,137)( 66,141)( 67,139)( 68,143)( 69,138)( 70,136)( 71,140)( 72,144)( 73,160)( 74,155)( 75,159)( 76,157)( 77,161)( 78,156)( 79,154)( 80,158)( 81,162)(163,169)(165,168)(167,170)(172,187)(173,182)(174,186)(175,184)(176,188)(177,183)(178,181)(179,185)(180,189)(190,208)(191,212)(192,216)(193,214)(194,209)(195,213)(196,211)(197,215)(198,210)(200,203)(201,207)(202,205)(217,229)(218,233)(219,228)(220,226)(221,230)(222,234)(223,232)(224,227)(225,231)(235,238)(236,242)(240,243);
s2 := Sym(243)!(  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28,186)( 29,185)( 30,184)( 31,188)( 32,187)( 33,189)( 34,181)( 35,183)( 36,182)( 37,177)( 38,176)( 39,175)( 40,179)( 41,178)( 42,180)( 43,172)( 44,174)( 45,173)( 46,168)( 47,167)( 48,166)( 49,170)( 50,169)( 51,171)( 52,163)( 53,165)( 54,164)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)( 64, 83)( 65, 82)( 66, 84)( 67, 85)( 68, 87)( 69, 86)( 70, 90)( 71, 89)( 72, 88)( 73,101)( 74,100)( 75,102)( 76,103)( 77,105)( 78,104)( 79,108)( 80,107)( 81,106)(109,229)(110,231)(111,230)(112,234)(113,233)(114,232)(115,227)(116,226)(117,228)(118,220)(119,222)(120,221)(121,225)(122,224)(123,223)(124,218)(125,217)(126,219)(127,238)(128,240)(129,239)(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,147)(137,146)(138,145)(139,149)(140,148)(141,150)(142,151)(143,153)(144,152)(154,156)(157,158)(161,162)(190,192)(193,194)(197,198)(199,210)(200,209)(201,208)(202,212)(203,211)(204,213)(205,214)(206,216)(207,215);
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*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle