Polytope of Type {9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1782)
Rank : 3
Schlafli Type : {9,18}
Number of vertices, edges, etc : 36, 324, 72
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 : {9,6}*432
   4-fold quotients : {9,18}*324
   9-fold quotients : {3,6}*144
   12-fold quotients : {9,6}*108
   27-fold quotients : {3,6}*48
   36-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {3,3}*24
   108-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope