Polytope of Type {6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1296h
if this polytope has a name.
Group : SmallGroup(1296,943)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,18}*648h
   3-fold quotients : {6,12}*432c
   4-fold quotients : {3,18}*324
   6-fold quotients : {6,6}*216c
   9-fold quotients : {6,12}*144c
   12-fold quotients : {3,6}*108
   18-fold quotients : {6,6}*72c
   27-fold quotients : {6,4}*48a
   36-fold quotients : {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {3,2}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope