Polytope of Type {18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,9}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1782)
Rank : 3
Schlafli Type : {18,9}
Number of vertices, edges, etc : 72, 324, 36
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,9}*432
   4-fold quotients : {18,9}*324
   9-fold quotients : {6,3}*144
   12-fold quotients : {6,9}*108
   27-fold quotients : {6,3}*48
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {3,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)( 77, 81)
( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 99,100)(101,105)(102,106)(103,108)(104,107)(109,225)(110,226)(111,228)
(112,227)(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)(119,220)
(120,219)(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)(127,236)
(128,235)(129,229)(130,230)(131,232)(132,231)(133,249)(134,250)(135,252)
(136,251)(137,245)(138,246)(139,248)(140,247)(141,241)(142,242)(143,244)
(144,243)(145,261)(146,262)(147,264)(148,263)(149,257)(150,258)(151,260)
(152,259)(153,253)(154,254)(155,256)(156,255)(157,273)(158,274)(159,276)
(160,275)(161,269)(162,270)(163,272)(164,271)(165,265)(166,266)(167,268)
(168,267)(169,285)(170,286)(171,288)(172,287)(173,281)(174,282)(175,284)
(176,283)(177,277)(178,278)(179,280)(180,279)(181,297)(182,298)(183,300)
(184,299)(185,293)(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)
(192,291)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)
(200,307)(201,301)(202,302)(203,304)(204,303)(205,321)(206,322)(207,324)
(208,323)(209,317)(210,318)(211,320)(212,319)(213,313)(214,314)(215,316)
(216,315);;
s1 := (  1,109)(  2,112)(  3,111)(  4,110)(  5,117)(  6,120)(  7,119)(  8,118)
(  9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)( 24,138)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)( 32,130)
( 33,125)( 34,128)( 35,127)( 36,126)( 37,205)( 38,208)( 39,207)( 40,206)
( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)( 48,210)
( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)( 56,202)
( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)( 64,182)
( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)( 72,186)
( 73,169)( 74,172)( 75,171)( 76,170)( 77,177)( 78,180)( 79,179)( 80,178)
( 81,173)( 82,176)( 83,175)( 84,174)( 85,157)( 86,160)( 87,159)( 88,158)
( 89,165)( 90,168)( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)( 96,162)
( 97,145)( 98,148)( 99,147)(100,146)(101,153)(102,156)(103,155)(104,154)
(105,149)(106,152)(107,151)(108,150)(217,225)(218,228)(219,227)(220,226)
(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)(235,247)
(236,246)(237,241)(238,244)(239,243)(240,242)(253,321)(254,324)(255,323)
(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)
(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)(271,307)
(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)(279,299)
(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)(287,291)
(288,290);;
s2 := (  1, 38)(  2, 37)(  3, 39)(  4, 40)(  5, 46)(  6, 45)(  7, 47)(  8, 48)
(  9, 42)( 10, 41)( 11, 43)( 12, 44)( 13, 62)( 14, 61)( 15, 63)( 16, 64)
( 17, 70)( 18, 69)( 19, 71)( 20, 72)( 21, 66)( 22, 65)( 23, 67)( 24, 68)
( 25, 50)( 26, 49)( 27, 51)( 28, 52)( 29, 58)( 30, 57)( 31, 59)( 32, 60)
( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 73, 98)( 74, 97)( 75, 99)( 76,100)
( 77,106)( 78,105)( 79,107)( 80,108)( 81,102)( 82,101)( 83,103)( 84,104)
( 85, 86)( 89, 94)( 90, 93)( 91, 95)( 92, 96)(109,262)(110,261)(111,263)
(112,264)(113,258)(114,257)(115,259)(116,260)(117,254)(118,253)(119,255)
(120,256)(121,286)(122,285)(123,287)(124,288)(125,282)(126,281)(127,283)
(128,284)(129,278)(130,277)(131,279)(132,280)(133,274)(134,273)(135,275)
(136,276)(137,270)(138,269)(139,271)(140,272)(141,266)(142,265)(143,267)
(144,268)(145,226)(146,225)(147,227)(148,228)(149,222)(150,221)(151,223)
(152,224)(153,218)(154,217)(155,219)(156,220)(157,250)(158,249)(159,251)
(160,252)(161,246)(162,245)(163,247)(164,248)(165,242)(166,241)(167,243)
(168,244)(169,238)(170,237)(171,239)(172,240)(173,234)(174,233)(175,235)
(176,236)(177,230)(178,229)(179,231)(180,232)(181,322)(182,321)(183,323)
(184,324)(185,318)(186,317)(187,319)(188,320)(189,314)(190,313)(191,315)
(192,316)(193,310)(194,309)(195,311)(196,312)(197,306)(198,305)(199,307)
(200,308)(201,302)(202,301)(203,303)(204,304)(205,298)(206,297)(207,299)
(208,300)(209,294)(210,293)(211,295)(212,296)(213,290)(214,289)(215,291)
(216,292);;
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*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(324)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)
( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 99,100)(101,105)(102,106)(103,108)(104,107)(109,225)(110,226)
(111,228)(112,227)(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)
(119,220)(120,219)(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)
(127,236)(128,235)(129,229)(130,230)(131,232)(132,231)(133,249)(134,250)
(135,252)(136,251)(137,245)(138,246)(139,248)(140,247)(141,241)(142,242)
(143,244)(144,243)(145,261)(146,262)(147,264)(148,263)(149,257)(150,258)
(151,260)(152,259)(153,253)(154,254)(155,256)(156,255)(157,273)(158,274)
(159,276)(160,275)(161,269)(162,270)(163,272)(164,271)(165,265)(166,266)
(167,268)(168,267)(169,285)(170,286)(171,288)(172,287)(173,281)(174,282)
(175,284)(176,283)(177,277)(178,278)(179,280)(180,279)(181,297)(182,298)
(183,300)(184,299)(185,293)(186,294)(187,296)(188,295)(189,289)(190,290)
(191,292)(192,291)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)
(199,308)(200,307)(201,301)(202,302)(203,304)(204,303)(205,321)(206,322)
(207,324)(208,323)(209,317)(210,318)(211,320)(212,319)(213,313)(214,314)
(215,316)(216,315);
s1 := Sym(324)!(  1,109)(  2,112)(  3,111)(  4,110)(  5,117)(  6,120)(  7,119)
(  8,118)(  9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)
( 24,138)( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)
( 32,130)( 33,125)( 34,128)( 35,127)( 36,126)( 37,205)( 38,208)( 39,207)
( 40,206)( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)
( 48,210)( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)
( 56,202)( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)
( 64,182)( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)
( 72,186)( 73,169)( 74,172)( 75,171)( 76,170)( 77,177)( 78,180)( 79,179)
( 80,178)( 81,173)( 82,176)( 83,175)( 84,174)( 85,157)( 86,160)( 87,159)
( 88,158)( 89,165)( 90,168)( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)
( 96,162)( 97,145)( 98,148)( 99,147)(100,146)(101,153)(102,156)(103,155)
(104,154)(105,149)(106,152)(107,151)(108,150)(217,225)(218,228)(219,227)
(220,226)(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)
(235,247)(236,246)(237,241)(238,244)(239,243)(240,242)(253,321)(254,324)
(255,323)(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)
(263,315)(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)
(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)
(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)
(287,291)(288,290);
s2 := Sym(324)!(  1, 38)(  2, 37)(  3, 39)(  4, 40)(  5, 46)(  6, 45)(  7, 47)
(  8, 48)(  9, 42)( 10, 41)( 11, 43)( 12, 44)( 13, 62)( 14, 61)( 15, 63)
( 16, 64)( 17, 70)( 18, 69)( 19, 71)( 20, 72)( 21, 66)( 22, 65)( 23, 67)
( 24, 68)( 25, 50)( 26, 49)( 27, 51)( 28, 52)( 29, 58)( 30, 57)( 31, 59)
( 32, 60)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 73, 98)( 74, 97)( 75, 99)
( 76,100)( 77,106)( 78,105)( 79,107)( 80,108)( 81,102)( 82,101)( 83,103)
( 84,104)( 85, 86)( 89, 94)( 90, 93)( 91, 95)( 92, 96)(109,262)(110,261)
(111,263)(112,264)(113,258)(114,257)(115,259)(116,260)(117,254)(118,253)
(119,255)(120,256)(121,286)(122,285)(123,287)(124,288)(125,282)(126,281)
(127,283)(128,284)(129,278)(130,277)(131,279)(132,280)(133,274)(134,273)
(135,275)(136,276)(137,270)(138,269)(139,271)(140,272)(141,266)(142,265)
(143,267)(144,268)(145,226)(146,225)(147,227)(148,228)(149,222)(150,221)
(151,223)(152,224)(153,218)(154,217)(155,219)(156,220)(157,250)(158,249)
(159,251)(160,252)(161,246)(162,245)(163,247)(164,248)(165,242)(166,241)
(167,243)(168,244)(169,238)(170,237)(171,239)(172,240)(173,234)(174,233)
(175,235)(176,236)(177,230)(178,229)(179,231)(180,232)(181,322)(182,321)
(183,323)(184,324)(185,318)(186,317)(187,319)(188,320)(189,314)(190,313)
(191,315)(192,316)(193,310)(194,309)(195,311)(196,312)(197,306)(198,305)
(199,307)(200,308)(201,302)(202,301)(203,303)(204,304)(205,298)(206,297)
(207,299)(208,300)(209,294)(210,293)(211,295)(212,296)(213,290)(214,289)
(215,291)(216,292);
poly := sub<Sym(324)|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*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope