Polytope of Type {9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*972e
if this polytope has a name.
Group : SmallGroup(972,105)
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 : {9,6}*324d
   9-fold quotients : {3,6}*108
   27-fold quotients : {3,6}*36
   81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,18}*1944o
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope