Polytope of Type {9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*972a
if this polytope has a name.
Group : SmallGroup(972,99)
Rank : 3
Schlafli Type : {9,18}
Number of vertices, edges, etc : 27, 243, 54
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,18,2} of size 1944
Vertex Figure Of :
   {2,9,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,18}*324, {9,6}*324a
   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 : {18,18}*1944a
Permutation Representation (GAP) :

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