Polytope of Type {12,4}

Play with this polytope as a twisty puzzle

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

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

Twisty Puzzle