Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,839)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s₀s₁s₂ : 12
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}*648d
   3-fold quotients : {12,6}*432a
   4-fold quotients : {9,6}*324b
   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, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)
( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)
( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)(113,140)
(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)(121,159)
(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)(129,146)
(130,150)(131,149)(132,148)(133,152)(134,151)(135,153)(163,244)(164,246)
(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)(172,262)
(173,264)(174,263)(175,267)(176,266)(177,265)(178,269)(179,268)(180,270)
(181,253)(182,255)(183,254)(184,258)(185,257)(186,256)(187,260)(188,259)
(189,261)(190,298)(191,300)(192,299)(193,303)(194,302)(195,301)(196,305)
(197,304)(198,306)(199,316)(200,318)(201,317)(202,321)(203,320)(204,319)
(205,323)(206,322)(207,324)(208,307)(209,309)(210,308)(211,312)(212,311)
(213,310)(214,314)(215,313)(216,315)(217,271)(218,273)(219,272)(220,276)
(221,275)(222,274)(223,278)(224,277)(225,279)(226,289)(227,291)(228,290)
(229,294)(230,293)(231,292)(232,296)(233,295)(234,297)(235,280)(236,282)
(237,281)(238,285)(239,284)(240,283)(241,287)(242,286)(243,288);;
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, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)(277,301)
(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)(285,314)
(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)(293,324)
(294,323)(295,319)(296,321)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*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,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)
( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)
( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)
( 39, 74)( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)
( 47, 66)( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)
( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)
( 96,103)( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)
(113,140)(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)
(121,159)(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)
(129,146)(130,150)(131,149)(132,148)(133,152)(134,151)(135,153)(163,244)
(164,246)(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)
(172,262)(173,264)(174,263)(175,267)(176,266)(177,265)(178,269)(179,268)
(180,270)(181,253)(182,255)(183,254)(184,258)(185,257)(186,256)(187,260)
(188,259)(189,261)(190,298)(191,300)(192,299)(193,303)(194,302)(195,301)
(196,305)(197,304)(198,306)(199,316)(200,318)(201,317)(202,321)(203,320)
(204,319)(205,323)(206,322)(207,324)(208,307)(209,309)(210,308)(211,312)
(212,311)(213,310)(214,314)(215,313)(216,315)(217,271)(218,273)(219,272)
(220,276)(221,275)(222,274)(223,278)(224,277)(225,279)(226,289)(227,291)
(228,290)(229,294)(230,293)(231,292)(232,296)(233,295)(234,297)(235,280)
(236,282)(237,281)(238,285)(239,284)(240,283)(241,287)(242,286)(243,288);
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, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)
(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)
(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)
(211,241)(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)
(277,301)(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)
(285,314)(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)
(293,324)(294,323)(295,319)(296,321)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*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