Polytope of Type {9,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*972e
if this polytope has a name.
Group : SmallGroup(972,111)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 81, 243, 54
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,6,2} of size 1944
Vertex Figure Of :
   {2,9,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,6}*324a, {9,6}*324c, {9,6}*324d, {3,6}*324
   9-fold quotients : {9,6}*108, {3,6}*108
   27-fold quotients : {9,2}*36, {3,6}*36
   81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*1944i
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {9}*18
      33 vertex figures:
         24 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {9,6}

Twisty Puzzle