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}*1944a
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 81, 486, 243
Order of s0s1s2 : 18
Order of s0s1s2s1 : 9
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   3-fold quotients : {4,4}*648
   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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      81 facets:
         81 of {4}*8
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*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*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*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*s2*s1*s0*s1*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 9.
      27 facets:
         27 of {4}*8
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      9 vertex figures:
         9 of {12}*24

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

Twisty Puzzle