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Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296h
if this polytope has a name.
Group : SmallGroup(1296,943)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s0s1s2 : 12
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) :
   2-fold quotients : {18,6}*648h
   3-fold quotients : {12,6}*432c
   4-fold quotients : {18,3}*324
   6-fold quotients : {6,6}*216a
   9-fold quotients : {12,6}*144c
   12-fold quotients : {6,3}*108
   18-fold quotients : {6,6}*72b
   27-fold quotients : {4,6}*48a
   36-fold quotients : {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  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);;
s1 := (  1,190)(  2,192)(  3,191)(  4,195)(  5,194)(  6,193)(  7,197)(  8,196)
(  9,198)( 10,215)( 11,214)( 12,216)( 13,208)( 14,210)( 15,209)( 16,213)
( 17,212)( 18,211)( 19,202)( 20,204)( 21,203)( 22,207)( 23,206)( 24,205)
( 25,200)( 26,199)( 27,201)( 28,163)( 29,165)( 30,164)( 31,168)( 32,167)
( 33,166)( 34,170)( 35,169)( 36,171)( 37,188)( 38,187)( 39,189)( 40,181)
( 41,183)( 42,182)( 43,186)( 44,185)( 45,184)( 46,175)( 47,177)( 48,176)
( 49,180)( 50,179)( 51,178)( 52,173)( 53,172)( 54,174)( 55,219)( 56,218)
( 57,217)( 58,221)( 59,220)( 60,222)( 61,223)( 62,225)( 63,224)( 64,241)
( 65,243)( 66,242)( 67,237)( 68,236)( 69,235)( 70,239)( 71,238)( 72,240)
( 73,231)( 74,230)( 75,229)( 76,233)( 77,232)( 78,234)( 79,226)( 80,228)
( 81,227)( 82,271)( 83,273)( 84,272)( 85,276)( 86,275)( 87,274)( 88,278)
( 89,277)( 90,279)( 91,296)( 92,295)( 93,297)( 94,289)( 95,291)( 96,290)
( 97,294)( 98,293)( 99,292)(100,283)(101,285)(102,284)(103,288)(104,287)
(105,286)(106,281)(107,280)(108,282)(109,244)(110,246)(111,245)(112,249)
(113,248)(114,247)(115,251)(116,250)(117,252)(118,269)(119,268)(120,270)
(121,262)(122,264)(123,263)(124,267)(125,266)(126,265)(127,256)(128,258)
(129,257)(130,261)(131,260)(132,259)(133,254)(134,253)(135,255)(136,300)
(137,299)(138,298)(139,302)(140,301)(141,303)(142,304)(143,306)(144,305)
(145,322)(146,324)(147,323)(148,318)(149,317)(150,316)(151,320)(152,319)
(153,321)(154,312)(155,311)(156,310)(157,314)(158,313)(159,315)(160,307)
(161,309)(162,308);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)
(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 66)( 29, 65)( 30, 64)( 31, 69)
( 32, 68)( 33, 67)( 34, 72)( 35, 71)( 36, 70)( 37, 57)( 38, 56)( 39, 55)
( 40, 60)( 41, 59)( 42, 58)( 43, 63)( 44, 62)( 45, 61)( 46, 75)( 47, 74)
( 48, 73)( 49, 78)( 50, 77)( 51, 76)( 52, 81)( 53, 80)( 54, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)
(101,102)(104,105)(107,108)(109,147)(110,146)(111,145)(112,150)(113,149)
(114,148)(115,153)(116,152)(117,151)(118,138)(119,137)(120,136)(121,141)
(122,140)(123,139)(124,144)(125,143)(126,142)(127,156)(128,155)(129,154)
(130,159)(131,158)(132,157)(133,162)(134,161)(135,160)(163,172)(164,174)
(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)(182,183)
(185,186)(188,189)(190,228)(191,227)(192,226)(193,231)(194,230)(195,229)
(196,234)(197,233)(198,232)(199,219)(200,218)(201,217)(202,222)(203,221)
(204,220)(205,225)(206,224)(207,223)(208,237)(209,236)(210,235)(211,240)
(212,239)(213,238)(214,243)(215,242)(216,241)(244,253)(245,255)(246,254)
(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)
(269,270)(271,309)(272,308)(273,307)(274,312)(275,311)(276,310)(277,315)
(278,314)(279,313)(280,300)(281,299)(282,298)(283,303)(284,302)(285,301)
(286,306)(287,305)(288,304)(289,318)(290,317)(291,316)(292,321)(293,320)
(294,319)(295,324)(296,323)(297,322);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(324)!(  1,190)(  2,192)(  3,191)(  4,195)(  5,194)(  6,193)(  7,197)
(  8,196)(  9,198)( 10,215)( 11,214)( 12,216)( 13,208)( 14,210)( 15,209)
( 16,213)( 17,212)( 18,211)( 19,202)( 20,204)( 21,203)( 22,207)( 23,206)
( 24,205)( 25,200)( 26,199)( 27,201)( 28,163)( 29,165)( 30,164)( 31,168)
( 32,167)( 33,166)( 34,170)( 35,169)( 36,171)( 37,188)( 38,187)( 39,189)
( 40,181)( 41,183)( 42,182)( 43,186)( 44,185)( 45,184)( 46,175)( 47,177)
( 48,176)( 49,180)( 50,179)( 51,178)( 52,173)( 53,172)( 54,174)( 55,219)
( 56,218)( 57,217)( 58,221)( 59,220)( 60,222)( 61,223)( 62,225)( 63,224)
( 64,241)( 65,243)( 66,242)( 67,237)( 68,236)( 69,235)( 70,239)( 71,238)
( 72,240)( 73,231)( 74,230)( 75,229)( 76,233)( 77,232)( 78,234)( 79,226)
( 80,228)( 81,227)( 82,271)( 83,273)( 84,272)( 85,276)( 86,275)( 87,274)
( 88,278)( 89,277)( 90,279)( 91,296)( 92,295)( 93,297)( 94,289)( 95,291)
( 96,290)( 97,294)( 98,293)( 99,292)(100,283)(101,285)(102,284)(103,288)
(104,287)(105,286)(106,281)(107,280)(108,282)(109,244)(110,246)(111,245)
(112,249)(113,248)(114,247)(115,251)(116,250)(117,252)(118,269)(119,268)
(120,270)(121,262)(122,264)(123,263)(124,267)(125,266)(126,265)(127,256)
(128,258)(129,257)(130,261)(131,260)(132,259)(133,254)(134,253)(135,255)
(136,300)(137,299)(138,298)(139,302)(140,301)(141,303)(142,304)(143,306)
(144,305)(145,322)(146,324)(147,323)(148,318)(149,317)(150,316)(151,320)
(152,319)(153,321)(154,312)(155,311)(156,310)(157,314)(158,313)(159,315)
(160,307)(161,309)(162,308);
s2 := Sym(324)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)
(  8, 18)(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 66)( 29, 65)( 30, 64)
( 31, 69)( 32, 68)( 33, 67)( 34, 72)( 35, 71)( 36, 70)( 37, 57)( 38, 56)
( 39, 55)( 40, 60)( 41, 59)( 42, 58)( 43, 63)( 44, 62)( 45, 61)( 46, 75)
( 47, 74)( 48, 73)( 49, 78)( 50, 77)( 51, 76)( 52, 81)( 53, 80)( 54, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)
( 90, 98)(101,102)(104,105)(107,108)(109,147)(110,146)(111,145)(112,150)
(113,149)(114,148)(115,153)(116,152)(117,151)(118,138)(119,137)(120,136)
(121,141)(122,140)(123,139)(124,144)(125,143)(126,142)(127,156)(128,155)
(129,154)(130,159)(131,158)(132,157)(133,162)(134,161)(135,160)(163,172)
(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)
(182,183)(185,186)(188,189)(190,228)(191,227)(192,226)(193,231)(194,230)
(195,229)(196,234)(197,233)(198,232)(199,219)(200,218)(201,217)(202,222)
(203,221)(204,220)(205,225)(206,224)(207,223)(208,237)(209,236)(210,235)
(211,240)(212,239)(213,238)(214,243)(215,242)(216,241)(244,253)(245,255)
(246,254)(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)
(266,267)(269,270)(271,309)(272,308)(273,307)(274,312)(275,311)(276,310)
(277,315)(278,314)(279,313)(280,300)(281,299)(282,298)(283,303)(284,302)
(285,301)(286,306)(287,305)(288,304)(289,318)(290,317)(291,316)(292,321)
(293,320)(294,319)(295,324)(296,323)(297,322);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope