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}*1944c
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 81, 486, 243
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 3
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   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*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*s2*s1*s2*s1*s0*s2*s1*s2*s1*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*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      81 facets:
         81 of {4}*8
      33 vertex figures:
         24 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*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*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 9.
      27 facets:
         27 of {4}*8
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      9 vertex figures:
         9 of {12}*24

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, 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);;
s1 := (  2,  3)(  4,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,184)( 29,186)( 30,185)( 31,189)( 32,188)( 33,187)( 34,182)( 35,181)( 36,183)( 37,179)( 38,178)( 39,180)( 40,172)( 41,174)( 42,173)( 43,177)( 44,176)( 45,175)( 46,165)( 47,164)( 48,163)( 49,167)( 50,166)( 51,168)( 52,169)( 53,171)( 54,170)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)( 64, 86)( 65, 85)( 66, 87)( 67, 88)( 68, 90)( 69, 89)( 70, 84)( 71, 83)( 72, 82)( 73,108)( 74,107)( 75,106)( 76,101)( 77,100)( 78,102)( 79,103)( 80,105)( 81,104)(109,226)(110,228)(111,227)(112,231)(113,230)(114,229)(115,233)(116,232)(117,234)(118,221)(119,220)(120,222)(121,223)(122,225)(123,224)(124,219)(125,218)(126,217)(127,243)(128,242)(129,241)(130,236)(131,235)(132,237)(133,238)(134,240)(135,239)(136,149)(137,148)(138,150)(139,151)(140,153)(141,152)(142,147)(143,146)(144,145)(155,156)(157,159)(160,161)(190,192)(193,194)(197,198)(199,211)(200,213)(201,212)(202,216)(203,215)(204,214)(205,209)(206,208)(207,210);;
s2 := (  1, 66)(  2, 67)(  3, 71)(  4, 72)(  5, 64)(  6, 68)(  7, 69)(  8, 70)(  9, 65)( 10, 61)( 11, 56)( 12, 60)( 13, 58)( 14, 62)( 15, 57)( 16, 55)( 17, 59)( 18, 63)( 19, 77)( 20, 81)( 21, 73)( 22, 74)( 23, 78)( 24, 79)( 25, 80)( 26, 75)( 27, 76)( 28, 41)( 29, 45)( 30, 37)( 31, 38)( 32, 42)( 33, 43)( 34, 44)( 35, 39)( 36, 40)( 46, 52)( 48, 51)( 50, 53)( 82,147)( 83,148)( 84,152)( 85,153)( 86,145)( 87,149)( 88,150)( 89,151)( 90,146)( 91,142)( 92,137)( 93,141)( 94,139)( 95,143)( 96,138)( 97,136)( 98,140)( 99,144)(100,158)(101,162)(102,154)(103,155)(104,159)(105,160)(106,161)(107,156)(108,157)(109,122)(110,126)(111,118)(112,119)(113,123)(114,124)(115,125)(116,120)(117,121)(127,133)(129,132)(131,134)(163,228)(164,229)(165,233)(166,234)(167,226)(168,230)(169,231)(170,232)(171,227)(172,223)(173,218)(174,222)(175,220)(176,224)(177,219)(178,217)(179,221)(180,225)(181,239)(182,243)(183,235)(184,236)(185,240)(186,241)(187,242)(188,237)(189,238)(190,203)(191,207)(192,199)(193,200)(194,204)(195,205)(196,206)(197,201)(198,202)(208,214)(210,213)(212,215);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1 ];;
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, 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);
s1 := Sym(243)!(  2,  3)(  4,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,184)( 29,186)( 30,185)( 31,189)( 32,188)( 33,187)( 34,182)( 35,181)( 36,183)( 37,179)( 38,178)( 39,180)( 40,172)( 41,174)( 42,173)( 43,177)( 44,176)( 45,175)( 46,165)( 47,164)( 48,163)( 49,167)( 50,166)( 51,168)( 52,169)( 53,171)( 54,170)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)( 64, 86)( 65, 85)( 66, 87)( 67, 88)( 68, 90)( 69, 89)( 70, 84)( 71, 83)( 72, 82)( 73,108)( 74,107)( 75,106)( 76,101)( 77,100)( 78,102)( 79,103)( 80,105)( 81,104)(109,226)(110,228)(111,227)(112,231)(113,230)(114,229)(115,233)(116,232)(117,234)(118,221)(119,220)(120,222)(121,223)(122,225)(123,224)(124,219)(125,218)(126,217)(127,243)(128,242)(129,241)(130,236)(131,235)(132,237)(133,238)(134,240)(135,239)(136,149)(137,148)(138,150)(139,151)(140,153)(141,152)(142,147)(143,146)(144,145)(155,156)(157,159)(160,161)(190,192)(193,194)(197,198)(199,211)(200,213)(201,212)(202,216)(203,215)(204,214)(205,209)(206,208)(207,210);
s2 := Sym(243)!(  1, 66)(  2, 67)(  3, 71)(  4, 72)(  5, 64)(  6, 68)(  7, 69)(  8, 70)(  9, 65)( 10, 61)( 11, 56)( 12, 60)( 13, 58)( 14, 62)( 15, 57)( 16, 55)( 17, 59)( 18, 63)( 19, 77)( 20, 81)( 21, 73)( 22, 74)( 23, 78)( 24, 79)( 25, 80)( 26, 75)( 27, 76)( 28, 41)( 29, 45)( 30, 37)( 31, 38)( 32, 42)( 33, 43)( 34, 44)( 35, 39)( 36, 40)( 46, 52)( 48, 51)( 50, 53)( 82,147)( 83,148)( 84,152)( 85,153)( 86,145)( 87,149)( 88,150)( 89,151)( 90,146)( 91,142)( 92,137)( 93,141)( 94,139)( 95,143)( 96,138)( 97,136)( 98,140)( 99,144)(100,158)(101,162)(102,154)(103,155)(104,159)(105,160)(106,161)(107,156)(108,157)(109,122)(110,126)(111,118)(112,119)(113,123)(114,124)(115,125)(116,120)(117,121)(127,133)(129,132)(131,134)(163,228)(164,229)(165,233)(166,234)(167,226)(168,230)(169,231)(170,232)(171,227)(172,223)(173,218)(174,222)(175,220)(176,224)(177,219)(178,217)(179,221)(180,225)(181,239)(182,243)(183,235)(184,236)(185,240)(186,241)(187,242)(188,237)(189,238)(190,203)(191,207)(192,199)(193,200)(194,204)(195,205)(196,206)(197,201)(198,202)(208,214)(210,213)(212,215);
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1 >;

References : None.
to this polytope

Polytope of Type {4,12}

Twisty Puzzle