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}*1944e
if this polytope has a name.
Group : SmallGroup(1944,806)
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
   Self-Petrie
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}*216b
   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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 3.
      27 facets:
         27 of {12}*24
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 3.
      27 facets:
         27 of {12}*24
      36 vertex figures:
         9 of {18}*36
         27 of {6}*12

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 := (  1, 11)(  2, 12)(  3, 10)(  4, 18)(  5, 16)(  6, 17)(  7, 13)(  8, 14)(  9, 15)( 22, 26)( 23, 27)( 24, 25)( 28, 90)( 29, 88)( 30, 89)( 31, 85)( 32, 86)( 33, 87)( 34, 83)( 35, 84)( 36, 82)( 37,106)( 38,107)( 39,108)( 40,104)( 41,105)( 42,103)( 43,102)( 44,100)( 45,101)( 46, 98)( 47, 99)( 48, 97)( 49, 96)( 50, 94)( 51, 95)( 52, 91)( 53, 92)( 54, 93)( 55,181)( 56,182)( 57,183)( 58,188)( 59,189)( 60,187)( 61,186)( 62,184)( 63,185)( 64,173)( 65,174)( 66,172)( 67,180)( 68,178)( 69,179)( 70,175)( 71,176)( 72,177)( 73,165)( 74,163)( 75,164)( 76,169)( 77,170)( 78,171)( 79,167)( 80,168)( 81,166)(112,116)(113,117)(114,115)(118,128)(119,129)(120,127)(121,135)(122,133)(123,134)(124,130)(125,131)(126,132)(136,194)(137,195)(138,193)(139,192)(140,190)(141,191)(142,196)(143,197)(144,198)(145,213)(146,211)(147,212)(148,208)(149,209)(150,210)(151,215)(152,216)(153,214)(154,202)(155,203)(156,204)(157,200)(158,201)(159,199)(160,207)(161,205)(162,206)(217,227)(218,228)(219,226)(220,234)(221,232)(222,233)(223,229)(224,230)(225,231)(238,242)(239,243)(240,241);;
s2 := (  1, 64)(  2, 66)(  3, 65)(  4, 70)(  5, 72)(  6, 71)(  7, 67)(  8, 69)(  9, 68)( 10, 78)( 11, 77)( 12, 76)( 13, 75)( 14, 74)( 15, 73)( 16, 81)( 17, 80)( 18, 79)( 19, 62)( 20, 61)( 21, 63)( 22, 59)( 23, 58)( 24, 60)( 25, 56)( 26, 55)( 27, 57)( 28, 35)( 29, 34)( 30, 36)( 31, 32)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 46, 51)( 47, 50)( 48, 49)( 52, 54)( 82,231)( 83,230)( 84,229)( 85,228)( 86,227)( 87,226)( 88,234)( 89,233)( 90,232)( 91,242)( 92,241)( 93,243)( 94,239)( 95,238)( 96,240)( 97,236)( 98,235)( 99,237)(100,217)(101,219)(102,218)(103,223)(104,225)(105,224)(106,220)(107,222)(108,221)(109,190)(110,192)(111,191)(112,196)(113,198)(114,197)(115,193)(116,195)(117,194)(118,204)(119,203)(120,202)(121,201)(122,200)(123,199)(124,207)(125,206)(126,205)(127,215)(128,214)(129,216)(130,212)(131,211)(132,213)(133,209)(134,208)(135,210)(136,181)(137,183)(138,182)(139,187)(140,189)(141,188)(142,184)(143,186)(144,185)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,171)(152,170)(153,169)(154,179)(155,178)(156,180)(157,176)(158,175)(159,177)(160,173)(161,172)(162,174);;
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*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0 ];;
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)!(  1, 11)(  2, 12)(  3, 10)(  4, 18)(  5, 16)(  6, 17)(  7, 13)(  8, 14)(  9, 15)( 22, 26)( 23, 27)( 24, 25)( 28, 90)( 29, 88)( 30, 89)( 31, 85)( 32, 86)( 33, 87)( 34, 83)( 35, 84)( 36, 82)( 37,106)( 38,107)( 39,108)( 40,104)( 41,105)( 42,103)( 43,102)( 44,100)( 45,101)( 46, 98)( 47, 99)( 48, 97)( 49, 96)( 50, 94)( 51, 95)( 52, 91)( 53, 92)( 54, 93)( 55,181)( 56,182)( 57,183)( 58,188)( 59,189)( 60,187)( 61,186)( 62,184)( 63,185)( 64,173)( 65,174)( 66,172)( 67,180)( 68,178)( 69,179)( 70,175)( 71,176)( 72,177)( 73,165)( 74,163)( 75,164)( 76,169)( 77,170)( 78,171)( 79,167)( 80,168)( 81,166)(112,116)(113,117)(114,115)(118,128)(119,129)(120,127)(121,135)(122,133)(123,134)(124,130)(125,131)(126,132)(136,194)(137,195)(138,193)(139,192)(140,190)(141,191)(142,196)(143,197)(144,198)(145,213)(146,211)(147,212)(148,208)(149,209)(150,210)(151,215)(152,216)(153,214)(154,202)(155,203)(156,204)(157,200)(158,201)(159,199)(160,207)(161,205)(162,206)(217,227)(218,228)(219,226)(220,234)(221,232)(222,233)(223,229)(224,230)(225,231)(238,242)(239,243)(240,241);
s2 := Sym(243)!(  1, 64)(  2, 66)(  3, 65)(  4, 70)(  5, 72)(  6, 71)(  7, 67)(  8, 69)(  9, 68)( 10, 78)( 11, 77)( 12, 76)( 13, 75)( 14, 74)( 15, 73)( 16, 81)( 17, 80)( 18, 79)( 19, 62)( 20, 61)( 21, 63)( 22, 59)( 23, 58)( 24, 60)( 25, 56)( 26, 55)( 27, 57)( 28, 35)( 29, 34)( 30, 36)( 31, 32)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 46, 51)( 47, 50)( 48, 49)( 52, 54)( 82,231)( 83,230)( 84,229)( 85,228)( 86,227)( 87,226)( 88,234)( 89,233)( 90,232)( 91,242)( 92,241)( 93,243)( 94,239)( 95,238)( 96,240)( 97,236)( 98,235)( 99,237)(100,217)(101,219)(102,218)(103,223)(104,225)(105,224)(106,220)(107,222)(108,221)(109,190)(110,192)(111,191)(112,196)(113,198)(114,197)(115,193)(116,195)(117,194)(118,204)(119,203)(120,202)(121,201)(122,200)(123,199)(124,207)(125,206)(126,205)(127,215)(128,214)(129,216)(130,212)(131,211)(132,213)(133,209)(134,208)(135,210)(136,181)(137,183)(138,182)(139,187)(140,189)(141,188)(142,184)(143,186)(144,185)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,171)(152,170)(153,169)(154,179)(155,178)(156,180)(157,176)(158,175)(159,177)(160,173)(161,172)(162,174);
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*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0 >;
References : None.
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