Polytope of Type {9,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,36}*1296
if this polytope has a name.
Group : SmallGroup(1296,1782)
Rank : 3
Schlafli Type : {9,36}
Number of vertices, edges, etc : 18, 324, 72
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 36
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 : {9,12}*432
   4-fold quotients : {9,18}*324
   9-fold quotients : {9,4}*144, {3,12}*144
   12-fold quotients : {9,6}*108
   18-fold quotients : {9,4}*72
   27-fold quotients : {3,4}*48
   36-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {3,4}*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)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 26)( 14, 25)
( 15, 28)( 16, 27)( 17, 30)( 18, 29)( 19, 32)( 20, 31)( 21, 34)( 22, 33)
( 23, 36)( 24, 35)( 37, 98)( 38, 97)( 39,100)( 40, 99)( 41,102)( 42,101)
( 43,104)( 44,103)( 45,106)( 46,105)( 47,108)( 48,107)( 49, 86)( 50, 85)
( 51, 88)( 52, 87)( 53, 90)( 54, 89)( 55, 92)( 56, 91)( 57, 94)( 58, 93)
( 59, 96)( 60, 95)( 61, 74)( 62, 73)( 63, 76)( 64, 75)( 65, 78)( 66, 77)
( 67, 80)( 68, 79)( 69, 82)( 70, 81)( 71, 84)( 72, 83)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,134)(122,133)(123,136)(124,135)
(125,138)(126,137)(127,140)(128,139)(129,142)(130,141)(131,144)(132,143)
(145,206)(146,205)(147,208)(148,207)(149,210)(150,209)(151,212)(152,211)
(153,214)(154,213)(155,216)(156,215)(157,194)(158,193)(159,196)(160,195)
(161,198)(162,197)(163,200)(164,199)(165,202)(166,201)(167,204)(168,203)
(169,182)(170,181)(171,184)(172,183)(173,186)(174,185)(175,188)(176,187)
(177,190)(178,189)(179,192)(180,191)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,242)(230,241)(231,244)(232,243)(233,246)(234,245)
(235,248)(236,247)(237,250)(238,249)(239,252)(240,251)(253,314)(254,313)
(255,316)(256,315)(257,318)(258,317)(259,320)(260,319)(261,322)(262,321)
(263,324)(264,323)(265,302)(266,301)(267,304)(268,303)(269,306)(270,305)
(271,308)(272,307)(273,310)(274,309)(275,312)(276,311)(277,290)(278,289)
(279,292)(280,291)(281,294)(282,293)(283,296)(284,295)(285,298)(286,297)
(287,300)(288,299);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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(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)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 26)
( 14, 25)( 15, 28)( 16, 27)( 17, 30)( 18, 29)( 19, 32)( 20, 31)( 21, 34)
( 22, 33)( 23, 36)( 24, 35)( 37, 98)( 38, 97)( 39,100)( 40, 99)( 41,102)
( 42,101)( 43,104)( 44,103)( 45,106)( 46,105)( 47,108)( 48,107)( 49, 86)
( 50, 85)( 51, 88)( 52, 87)( 53, 90)( 54, 89)( 55, 92)( 56, 91)( 57, 94)
( 58, 93)( 59, 96)( 60, 95)( 61, 74)( 62, 73)( 63, 76)( 64, 75)( 65, 78)
( 66, 77)( 67, 80)( 68, 79)( 69, 82)( 70, 81)( 71, 84)( 72, 83)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,134)(122,133)(123,136)
(124,135)(125,138)(126,137)(127,140)(128,139)(129,142)(130,141)(131,144)
(132,143)(145,206)(146,205)(147,208)(148,207)(149,210)(150,209)(151,212)
(152,211)(153,214)(154,213)(155,216)(156,215)(157,194)(158,193)(159,196)
(160,195)(161,198)(162,197)(163,200)(164,199)(165,202)(166,201)(167,204)
(168,203)(169,182)(170,181)(171,184)(172,183)(173,186)(174,185)(175,188)
(176,187)(177,190)(178,189)(179,192)(180,191)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,242)(230,241)(231,244)(232,243)(233,246)
(234,245)(235,248)(236,247)(237,250)(238,249)(239,252)(240,251)(253,314)
(254,313)(255,316)(256,315)(257,318)(258,317)(259,320)(260,319)(261,322)
(262,321)(263,324)(264,323)(265,302)(266,301)(267,304)(268,303)(269,306)
(270,305)(271,308)(272,307)(273,310)(274,309)(275,312)(276,311)(277,290)
(278,289)(279,292)(280,291)(281,294)(282,293)(283,296)(284,295)(285,298)
(286,297)(287,300)(288,299);
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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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