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}*1944d
if this polytope has a name.
Group : SmallGroup(1944,806)
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
   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=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> 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=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 3.
      81 facets:
         81 of {4}*8
      33 vertex figures:
         24 of {12}*24
         9 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 9.
      27 facets:
         27 of {4}*8
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      15 vertex figures:
         6 of {12}*24
         9 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, 42)( 29, 38)( 30, 43)( 31, 45)( 32, 41)( 33, 37)( 34, 39)( 35, 44)( 36, 40)( 46, 51)( 48, 52)( 49, 54)( 55, 73)( 56, 81)( 57, 77)( 58, 76)( 59, 75)( 60, 80)( 61, 79)( 62, 78)( 63, 74)( 65, 72)( 66, 68)( 69, 71)( 82,168)( 83,164)( 84,169)( 85,171)( 86,167)( 87,163)( 88,165)( 89,170)( 90,166)( 91,186)( 92,182)( 93,187)( 94,189)( 95,185)( 96,181)( 97,183)( 98,188)( 99,184)(100,177)(101,173)(102,178)(103,180)(104,176)(105,172)(106,174)(107,179)(108,175)(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)(136,240)(137,236)(138,241)(139,243)(140,239)(141,235)(142,237)(143,242)(144,238)(145,231)(146,227)(147,232)(148,234)(149,230)(150,226)(151,228)(152,233)(153,229)(154,222)(155,218)(156,223)(157,225)(158,221)(159,217)(160,219)(161,224)(162,220);;
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)( 28,181)( 29,183)( 30,182)( 31,186)( 32,185)( 33,184)( 34,188)( 35,187)( 36,189)( 37,176)( 38,175)( 39,177)( 40,178)( 41,180)( 42,179)( 43,174)( 44,173)( 45,172)( 46,171)( 47,170)( 48,169)( 49,164)( 50,163)( 51,165)( 52,166)( 53,168)( 54,167)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)( 64, 87)( 65, 86)( 66, 85)( 67, 89)( 68, 88)( 69, 90)( 70, 82)( 71, 84)( 72, 83)( 73,106)( 74,108)( 75,107)( 76,102)( 77,101)( 78,100)( 79,104)( 80,103)( 81,105)(109,233)(110,232)(111,234)(112,226)(113,228)(114,227)(115,231)(116,230)(117,229)(118,219)(119,218)(120,217)(121,221)(122,220)(123,222)(124,223)(125,225)(126,224)(127,238)(128,240)(129,239)(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,148)(137,150)(138,149)(139,153)(140,152)(141,151)(142,146)(143,145)(144,147)(154,156)(157,158)(161,162)(190,191)(194,195)(196,198)(199,213)(200,212)(201,211)(202,215)(203,214)(204,216)(205,208)(206,210)(207,209);;
s2 := (  1, 64)(  2, 68)(  3, 72)(  4, 70)(  5, 65)(  6, 69)(  7, 67)(  8, 71)(  9, 66)( 10, 62)( 11, 57)( 12, 58)( 13, 59)( 14, 63)( 15, 55)( 16, 56)( 17, 60)( 18, 61)( 19, 78)( 20, 79)( 21, 74)( 22, 75)( 23, 76)( 24, 80)( 25, 81)( 26, 73)( 27, 77)( 28, 44)( 29, 39)( 30, 40)( 31, 41)( 32, 45)( 33, 37)( 34, 38)( 35, 42)( 36, 43)( 47, 50)( 48, 54)( 49, 52)( 82,145)( 83,149)( 84,153)( 85,151)( 86,146)( 87,150)( 88,148)( 89,152)( 90,147)( 91,143)( 92,138)( 93,139)( 94,140)( 95,144)( 96,136)( 97,137)( 98,141)( 99,142)(100,159)(101,160)(102,155)(103,156)(104,157)(105,161)(106,162)(107,154)(108,158)(109,125)(110,120)(111,121)(112,122)(113,126)(114,118)(115,119)(116,123)(117,124)(128,131)(129,135)(130,133)(163,226)(164,230)(165,234)(166,232)(167,227)(168,231)(169,229)(170,233)(171,228)(172,224)(173,219)(174,220)(175,221)(176,225)(177,217)(178,218)(179,222)(180,223)(181,240)(182,241)(183,236)(184,237)(185,238)(186,242)(187,243)(188,235)(189,239)(190,206)(191,201)(192,202)(193,203)(194,207)(195,199)(196,200)(197,204)(198,205)(209,212)(210,216)(211,214);;
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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, 42)( 29, 38)( 30, 43)( 31, 45)( 32, 41)( 33, 37)( 34, 39)( 35, 44)( 36, 40)( 46, 51)( 48, 52)( 49, 54)( 55, 73)( 56, 81)( 57, 77)( 58, 76)( 59, 75)( 60, 80)( 61, 79)( 62, 78)( 63, 74)( 65, 72)( 66, 68)( 69, 71)( 82,168)( 83,164)( 84,169)( 85,171)( 86,167)( 87,163)( 88,165)( 89,170)( 90,166)( 91,186)( 92,182)( 93,187)( 94,189)( 95,185)( 96,181)( 97,183)( 98,188)( 99,184)(100,177)(101,173)(102,178)(103,180)(104,176)(105,172)(106,174)(107,179)(108,175)(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)(136,240)(137,236)(138,241)(139,243)(140,239)(141,235)(142,237)(143,242)(144,238)(145,231)(146,227)(147,232)(148,234)(149,230)(150,226)(151,228)(152,233)(153,229)(154,222)(155,218)(156,223)(157,225)(158,221)(159,217)(160,219)(161,224)(162,220);
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)( 28,181)( 29,183)( 30,182)( 31,186)( 32,185)( 33,184)( 34,188)( 35,187)( 36,189)( 37,176)( 38,175)( 39,177)( 40,178)( 41,180)( 42,179)( 43,174)( 44,173)( 45,172)( 46,171)( 47,170)( 48,169)( 49,164)( 50,163)( 51,165)( 52,166)( 53,168)( 54,167)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)( 64, 87)( 65, 86)( 66, 85)( 67, 89)( 68, 88)( 69, 90)( 70, 82)( 71, 84)( 72, 83)( 73,106)( 74,108)( 75,107)( 76,102)( 77,101)( 78,100)( 79,104)( 80,103)( 81,105)(109,233)(110,232)(111,234)(112,226)(113,228)(114,227)(115,231)(116,230)(117,229)(118,219)(119,218)(120,217)(121,221)(122,220)(123,222)(124,223)(125,225)(126,224)(127,238)(128,240)(129,239)(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,148)(137,150)(138,149)(139,153)(140,152)(141,151)(142,146)(143,145)(144,147)(154,156)(157,158)(161,162)(190,191)(194,195)(196,198)(199,213)(200,212)(201,211)(202,215)(203,214)(204,216)(205,208)(206,210)(207,209);
s2 := Sym(243)!(  1, 64)(  2, 68)(  3, 72)(  4, 70)(  5, 65)(  6, 69)(  7, 67)(  8, 71)(  9, 66)( 10, 62)( 11, 57)( 12, 58)( 13, 59)( 14, 63)( 15, 55)( 16, 56)( 17, 60)( 18, 61)( 19, 78)( 20, 79)( 21, 74)( 22, 75)( 23, 76)( 24, 80)( 25, 81)( 26, 73)( 27, 77)( 28, 44)( 29, 39)( 30, 40)( 31, 41)( 32, 45)( 33, 37)( 34, 38)( 35, 42)( 36, 43)( 47, 50)( 48, 54)( 49, 52)( 82,145)( 83,149)( 84,153)( 85,151)( 86,146)( 87,150)( 88,148)( 89,152)( 90,147)( 91,143)( 92,138)( 93,139)( 94,140)( 95,144)( 96,136)( 97,137)( 98,141)( 99,142)(100,159)(101,160)(102,155)(103,156)(104,157)(105,161)(106,162)(107,154)(108,158)(109,125)(110,120)(111,121)(112,122)(113,126)(114,118)(115,119)(116,123)(117,124)(128,131)(129,135)(130,133)(163,226)(164,230)(165,234)(166,232)(167,227)(168,231)(169,229)(170,233)(171,228)(172,224)(173,219)(174,220)(175,221)(176,225)(177,217)(178,218)(179,222)(180,223)(181,240)(182,241)(183,236)(184,237)(185,238)(186,242)(187,243)(188,235)(189,239)(190,206)(191,201)(192,202)(193,203)(194,207)(195,199)(196,200)(197,204)(198,205)(209,212)(210,216)(211,214);
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1 >;

References : None.
to this polytope

Polytope of Type {4,12}

Twisty Puzzle