Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,812)
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}*648a
   3-fold quotients : {36,6}*432b, {12,6}*432a
   4-fold quotients : {9,6}*324a
   6-fold quotients : {18,6}*216b, {6,6}*216c
   9-fold quotients : {36,2}*144, {12,6}*144b
   12-fold quotients : {9,6}*108, {3,6}*108
   18-fold quotients : {18,2}*72, {6,6}*72c
   27-fold quotients : {12,2}*48
   36-fold quotients : {9,2}*36, {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 := (  4,  7)(  5,  8)(  6,  9)( 10, 25)( 11, 26)( 12, 27)( 13, 22)( 14, 23)
( 15, 24)( 16, 19)( 17, 20)( 18, 21)( 28, 55)( 29, 56)( 30, 57)( 31, 61)
( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 79)( 38, 80)( 39, 81)
( 40, 76)( 41, 77)( 42, 78)( 43, 73)( 44, 74)( 45, 75)( 46, 70)( 47, 71)
( 48, 72)( 49, 67)( 50, 68)( 51, 69)( 52, 64)( 53, 65)( 54, 66)( 85, 88)
( 86, 89)( 87, 90)( 91,106)( 92,107)( 93,108)( 94,103)( 95,104)( 96,105)
( 97,100)( 98,101)( 99,102)(109,136)(110,137)(111,138)(112,142)(113,143)
(114,144)(115,139)(116,140)(117,141)(118,160)(119,161)(120,162)(121,157)
(122,158)(123,159)(124,154)(125,155)(126,156)(127,151)(128,152)(129,153)
(130,148)(131,149)(132,150)(133,145)(134,146)(135,147)(163,244)(164,245)
(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)(171,249)(172,268)
(173,269)(174,270)(175,265)(176,266)(177,267)(178,262)(179,263)(180,264)
(181,259)(182,260)(183,261)(184,256)(185,257)(186,258)(187,253)(188,254)
(189,255)(190,298)(191,299)(192,300)(193,304)(194,305)(195,306)(196,301)
(197,302)(198,303)(199,322)(200,323)(201,324)(202,319)(203,320)(204,321)
(205,316)(206,317)(207,318)(208,313)(209,314)(210,315)(211,310)(212,311)
(213,312)(214,307)(215,308)(216,309)(217,271)(218,272)(219,273)(220,277)
(221,278)(222,279)(223,274)(224,275)(225,276)(226,295)(227,296)(228,297)
(229,292)(230,293)(231,294)(232,289)(233,290)(234,291)(235,286)(236,287)
(237,288)(238,283)(239,284)(240,285)(241,280)(242,281)(243,282);;
s1 := (  1,227)(  2,228)(  3,226)(  4,233)(  5,234)(  6,232)(  7,230)(  8,231)
(  9,229)( 10,219)( 11,217)( 12,218)( 13,225)( 14,223)( 15,224)( 16,222)
( 17,220)( 18,221)( 19,241)( 20,242)( 21,243)( 22,238)( 23,239)( 24,240)
( 25,235)( 26,236)( 27,237)( 28,200)( 29,201)( 30,199)( 31,206)( 32,207)
( 33,205)( 34,203)( 35,204)( 36,202)( 37,192)( 38,190)( 39,191)( 40,198)
( 41,196)( 42,197)( 43,195)( 44,193)( 45,194)( 46,214)( 47,215)( 48,216)
( 49,211)( 50,212)( 51,213)( 52,208)( 53,209)( 54,210)( 55,173)( 56,174)
( 57,172)( 58,179)( 59,180)( 60,178)( 61,176)( 62,177)( 63,175)( 64,165)
( 65,163)( 66,164)( 67,171)( 68,169)( 69,170)( 70,168)( 71,166)( 72,167)
( 73,187)( 74,188)( 75,189)( 76,184)( 77,185)( 78,186)( 79,181)( 80,182)
( 81,183)( 82,308)( 83,309)( 84,307)( 85,314)( 86,315)( 87,313)( 88,311)
( 89,312)( 90,310)( 91,300)( 92,298)( 93,299)( 94,306)( 95,304)( 96,305)
( 97,303)( 98,301)( 99,302)(100,322)(101,323)(102,324)(103,319)(104,320)
(105,321)(106,316)(107,317)(108,318)(109,281)(110,282)(111,280)(112,287)
(113,288)(114,286)(115,284)(116,285)(117,283)(118,273)(119,271)(120,272)
(121,279)(122,277)(123,278)(124,276)(125,274)(126,275)(127,295)(128,296)
(129,297)(130,292)(131,293)(132,294)(133,289)(134,290)(135,291)(136,254)
(137,255)(138,253)(139,260)(140,261)(141,259)(142,257)(143,258)(144,256)
(145,246)(146,244)(147,245)(148,252)(149,250)(150,251)(151,249)(152,247)
(153,248)(154,268)(155,269)(156,270)(157,265)(158,266)(159,267)(160,262)
(161,263)(162,264);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)( 34, 61)
( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)( 42, 68)
( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)( 50, 78)
( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)(111,137)
(112,139)(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)(119,147)
(120,146)(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)(127,154)
(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161)
(164,165)(167,168)(170,171)(173,174)(176,177)(179,180)(182,183)(185,186)
(188,189)(190,217)(191,219)(192,218)(193,220)(194,222)(195,221)(196,223)
(197,225)(198,224)(199,226)(200,228)(201,227)(202,229)(203,231)(204,230)
(205,232)(206,234)(207,233)(208,235)(209,237)(210,236)(211,238)(212,240)
(213,239)(214,241)(215,243)(216,242)(245,246)(248,249)(251,252)(254,255)
(257,258)(260,261)(263,264)(266,267)(269,270)(271,298)(272,300)(273,299)
(274,301)(275,303)(276,302)(277,304)(278,306)(279,305)(280,307)(281,309)
(282,308)(283,310)(284,312)(285,311)(286,313)(287,315)(288,314)(289,316)
(290,318)(291,317)(292,319)(293,321)(294,320)(295,322)(296,324)(297,323);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope