Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296e
if this polytope has a name.
Group : SmallGroup(1296,854)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
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}*648e
   3-fold quotients : {12,6}*432a
   4-fold quotients : {9,6}*324d
   6-fold quotients : {6,6}*216c
   9-fold quotients : {12,6}*144b
   12-fold quotients : {3,6}*108
   18-fold quotients : {6,6}*72c
   27-fold quotients : {12,2}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*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, 21)( 11, 20)( 12, 19)( 13, 23)( 14, 22)
( 15, 24)( 16, 25)( 17, 27)( 18, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 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)( 91,102)( 92,101)( 93,100)( 94,104)( 95,103)( 96,105)
( 97,106)( 98,108)( 99,107)(109,136)(110,138)(111,137)(112,141)(113,140)
(114,139)(115,143)(116,142)(117,144)(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)(163,244)(164,246)
(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)(172,264)
(173,263)(174,262)(175,266)(176,265)(177,267)(178,268)(179,270)(180,269)
(181,255)(182,254)(183,253)(184,257)(185,256)(186,258)(187,259)(188,261)
(189,260)(190,298)(191,300)(192,299)(193,303)(194,302)(195,301)(196,305)
(197,304)(198,306)(199,318)(200,317)(201,316)(202,320)(203,319)(204,321)
(205,322)(206,324)(207,323)(208,309)(209,308)(210,307)(211,311)(212,310)
(213,312)(214,313)(215,315)(216,314)(217,271)(218,273)(219,272)(220,276)
(221,275)(222,274)(223,278)(224,277)(225,279)(226,291)(227,290)(228,289)
(229,293)(230,292)(231,294)(232,295)(233,297)(234,296)(235,282)(236,281)
(237,280)(238,284)(239,283)(240,285)(241,286)(242,288)(243,287);;
s1 := (  1,229)(  2,231)(  3,230)(  4,233)(  5,232)(  6,234)(  7,228)(  8,227)
(  9,226)( 10,225)( 11,224)( 12,223)( 13,217)( 14,219)( 15,218)( 16,221)
( 17,220)( 18,222)( 19,235)( 20,237)( 21,236)( 22,239)( 23,238)( 24,240)
( 25,243)( 26,242)( 27,241)( 28,202)( 29,204)( 30,203)( 31,206)( 32,205)
( 33,207)( 34,201)( 35,200)( 36,199)( 37,198)( 38,197)( 39,196)( 40,190)
( 41,192)( 42,191)( 43,194)( 44,193)( 45,195)( 46,208)( 47,210)( 48,209)
( 49,212)( 50,211)( 51,213)( 52,216)( 53,215)( 54,214)( 55,175)( 56,177)
( 57,176)( 58,179)( 59,178)( 60,180)( 61,174)( 62,173)( 63,172)( 64,171)
( 65,170)( 66,169)( 67,163)( 68,165)( 69,164)( 70,167)( 71,166)( 72,168)
( 73,181)( 74,183)( 75,182)( 76,185)( 77,184)( 78,186)( 79,189)( 80,188)
( 81,187)( 82,310)( 83,312)( 84,311)( 85,314)( 86,313)( 87,315)( 88,309)
( 89,308)( 90,307)( 91,306)( 92,305)( 93,304)( 94,298)( 95,300)( 96,299)
( 97,302)( 98,301)( 99,303)(100,316)(101,318)(102,317)(103,320)(104,319)
(105,321)(106,324)(107,323)(108,322)(109,283)(110,285)(111,284)(112,287)
(113,286)(114,288)(115,282)(116,281)(117,280)(118,279)(119,278)(120,277)
(121,271)(122,273)(123,272)(124,275)(125,274)(126,276)(127,289)(128,291)
(129,290)(130,293)(131,292)(132,294)(133,297)(134,296)(135,295)(136,256)
(137,258)(138,257)(139,260)(140,259)(141,261)(142,255)(143,254)(144,253)
(145,252)(146,251)(147,250)(148,244)(149,246)(150,245)(151,248)(152,247)
(153,249)(154,262)(155,264)(156,263)(157,266)(158,265)(159,267)(160,270)
(161,269)(162,268);;
s2 := (  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)
( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)
( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)
( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)
( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)
( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)
(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)
(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)
(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)
(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)
(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)
(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)
(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)
(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)
(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)
(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)
(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)
(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope