Polytope of Type {9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*972d
if this polytope has a name.
Group : SmallGroup(972,108)
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}*324b
   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}*1944h
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)(  4,197)(  5,196)(  6,198)(  7,192)(  8,191)
(  9,190)( 10,211)( 11,213)( 12,212)( 13,215)( 14,214)( 15,216)( 16,210)
( 17,209)( 18,208)( 19,202)( 20,204)( 21,203)( 22,206)( 23,205)( 24,207)
( 25,201)( 26,200)( 27,199)( 28,171)( 29,170)( 30,169)( 31,163)( 32,165)
( 33,164)( 34,167)( 35,166)( 36,168)( 37,189)( 38,188)( 39,187)( 40,181)
( 41,183)( 42,182)( 43,185)( 44,184)( 45,186)( 46,180)( 47,179)( 48,178)
( 49,172)( 50,174)( 51,173)( 52,176)( 53,175)( 54,177)( 55,218)( 56,217)
( 57,219)( 58,222)( 59,221)( 60,220)( 61,223)( 62,225)( 63,224)( 64,236)
( 65,235)( 66,237)( 67,240)( 68,239)( 69,238)( 70,241)( 71,243)( 72,242)
( 73,227)( 74,226)( 75,228)( 76,231)( 77,230)( 78,229)( 79,232)( 80,234)
( 81,233)( 82,112)( 83,114)( 84,113)( 85,116)( 86,115)( 87,117)( 88,111)
( 89,110)( 90,109)( 91,130)( 92,132)( 93,131)( 94,134)( 95,133)( 96,135)
( 97,129)( 98,128)( 99,127)(100,121)(101,123)(102,122)(103,125)(104,124)
(105,126)(106,120)(107,119)(108,118)(136,137)(139,141)(143,144)(145,155)
(146,154)(147,156)(148,159)(149,158)(150,157)(151,160)(152,162)(153,161);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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)(  4,197)(  5,196)(  6,198)(  7,192)
(  8,191)(  9,190)( 10,211)( 11,213)( 12,212)( 13,215)( 14,214)( 15,216)
( 16,210)( 17,209)( 18,208)( 19,202)( 20,204)( 21,203)( 22,206)( 23,205)
( 24,207)( 25,201)( 26,200)( 27,199)( 28,171)( 29,170)( 30,169)( 31,163)
( 32,165)( 33,164)( 34,167)( 35,166)( 36,168)( 37,189)( 38,188)( 39,187)
( 40,181)( 41,183)( 42,182)( 43,185)( 44,184)( 45,186)( 46,180)( 47,179)
( 48,178)( 49,172)( 50,174)( 51,173)( 52,176)( 53,175)( 54,177)( 55,218)
( 56,217)( 57,219)( 58,222)( 59,221)( 60,220)( 61,223)( 62,225)( 63,224)
( 64,236)( 65,235)( 66,237)( 67,240)( 68,239)( 69,238)( 70,241)( 71,243)
( 72,242)( 73,227)( 74,226)( 75,228)( 76,231)( 77,230)( 78,229)( 79,232)
( 80,234)( 81,233)( 82,112)( 83,114)( 84,113)( 85,116)( 86,115)( 87,117)
( 88,111)( 89,110)( 90,109)( 91,130)( 92,132)( 93,131)( 94,134)( 95,133)
( 96,135)( 97,129)( 98,128)( 99,127)(100,121)(101,123)(102,122)(103,125)
(104,124)(105,126)(106,120)(107,119)(108,118)(136,137)(139,141)(143,144)
(145,155)(146,154)(147,156)(148,159)(149,158)(150,157)(151,160)(152,162)
(153,161);
s2 := Sym(243)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)
(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)
(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)
(211,241)(212,243)(213,242)(214,238)(215,240)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope