Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296d
if this polytope has a name.
Group : SmallGroup(1296,840)
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}*648c
   3-fold quotients : {12,6}*432a
   4-fold quotients : {9,6}*324c
   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)(  4,233)(  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)( 19,236)( 20,235)( 21,237)( 22,240)( 23,239)( 24,238)
( 25,241)( 26,243)( 27,242)( 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)( 46,209)( 47,208)( 48,210)
( 49,213)( 50,212)( 51,211)( 52,214)( 53,216)( 54,215)( 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)
( 73,182)( 74,181)( 75,183)( 76,186)( 77,185)( 78,184)( 79,187)( 80,189)
( 81,188)( 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)(100,317)(101,316)(102,318)(103,321)(104,320)
(105,319)(106,322)(107,324)(108,323)(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)(127,290)(128,289)
(129,291)(130,294)(131,293)(132,292)(133,295)(134,297)(135,296)(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)(154,263)(155,262)(156,264)(157,267)(158,266)(159,265)(160,268)
(161,270)(162,269);;
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, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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)(  4,233)(  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)( 19,236)( 20,235)( 21,237)( 22,240)( 23,239)
( 24,238)( 25,241)( 26,243)( 27,242)( 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)( 46,209)( 47,208)
( 48,210)( 49,213)( 50,212)( 51,211)( 52,214)( 53,216)( 54,215)( 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)( 73,182)( 74,181)( 75,183)( 76,186)( 77,185)( 78,184)( 79,187)
( 80,189)( 81,188)( 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)(100,317)(101,316)(102,318)(103,321)
(104,320)(105,319)(106,322)(107,324)(108,323)(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)(127,290)
(128,289)(129,291)(130,294)(131,293)(132,292)(133,295)(134,297)(135,296)
(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)(154,263)(155,262)(156,264)(157,267)(158,266)(159,265)
(160,268)(161,270)(162,269);
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, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope