Polytope of Type {12,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1944i
if this polytope has a name.
Group : SmallGroup(1944,2325)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 54, 486, 81
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   9-fold quotients : {12,6}*216c
   27-fold quotients : {4,6}*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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      27 facets:
         27 of {12}*24
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      27 facets:
         27 of {12}*24
      36 vertex figures:
         27 of {6}*12
         9 of {18}*36

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {12,18}

Twisty Puzzle