Polytope of Type {4,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*1944
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 162, 486, 243
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {4,6}*216
   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*s2*s1*s0*s1*s2*s1> of order 3.
      81 facets:
         81 of {4}*8
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 3.
      81 facets:
         81 of {4}*8
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      81 facets:
         81 of {4}*8
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      81 facets:
         81 of {4}*8
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 3.
      81 facets:
         81 of {4}*8
      60 vertex figures:
         51 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 9.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0> of order 9.
      27 facets:
         27 of {4}*8
      24 vertex figures:
         15 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 27.
      9 facets:
         9 of {4}*8
      10 vertex figures:
         4 of {6}*12
         6 of {2}*4

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {4,6}

Twisty Puzzle