Polytope of Type {6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1296a
if this polytope has a name.
Group : SmallGroup(1296,812)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,18}*648a
   3-fold quotients : {6,36}*432b, {6,12}*432a
   4-fold quotients : {6,9}*324a
   6-fold quotients : {6,18}*216b, {6,6}*216a
   9-fold quotients : {2,36}*144, {6,12}*144b
   12-fold quotients : {6,9}*108, {6,3}*108
   18-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {2,12}*48
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {2,4}*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)(  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);;
s1 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)
(  9, 33)( 10, 54)( 11, 52)( 12, 53)( 13, 51)( 14, 49)( 15, 50)( 16, 48)
( 17, 46)( 18, 47)( 19, 44)( 20, 45)( 21, 43)( 22, 41)( 23, 42)( 24, 40)
( 25, 38)( 26, 39)( 27, 37)( 58, 61)( 59, 62)( 60, 63)( 64, 81)( 65, 79)
( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)( 82,109)
( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)
( 91,135)( 92,133)( 93,134)( 94,132)( 95,130)( 96,131)( 97,129)( 98,127)
( 99,128)(100,125)(101,126)(102,124)(103,122)(104,123)(105,121)(106,119)
(107,120)(108,118)(139,142)(140,143)(141,144)(145,162)(146,160)(147,161)
(148,159)(149,157)(150,158)(151,156)(152,154)(153,155)(163,271)(164,272)
(165,273)(166,277)(167,278)(168,279)(169,274)(170,275)(171,276)(172,297)
(173,295)(174,296)(175,294)(176,292)(177,293)(178,291)(179,289)(180,290)
(181,287)(182,288)(183,286)(184,284)(185,285)(186,283)(187,281)(188,282)
(189,280)(190,244)(191,245)(192,246)(193,250)(194,251)(195,252)(196,247)
(197,248)(198,249)(199,270)(200,268)(201,269)(202,267)(203,265)(204,266)
(205,264)(206,262)(207,263)(208,260)(209,261)(210,259)(211,257)(212,258)
(213,256)(214,254)(215,255)(216,253)(217,298)(218,299)(219,300)(220,304)
(221,305)(222,306)(223,301)(224,302)(225,303)(226,324)(227,322)(228,323)
(229,321)(230,319)(231,320)(232,318)(233,316)(234,317)(235,314)(236,315)
(237,313)(238,311)(239,312)(240,310)(241,308)(242,309)(243,307);;
s2 := (  1,172)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)(  8,176)
(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)( 16,166)
( 17,167)( 18,168)( 19,187)( 20,188)( 21,189)( 22,184)( 23,185)( 24,186)
( 25,181)( 26,182)( 27,183)( 28,226)( 29,227)( 30,228)( 31,232)( 32,233)
( 33,234)( 34,229)( 35,230)( 36,231)( 37,217)( 38,218)( 39,219)( 40,223)
( 41,224)( 42,225)( 43,220)( 44,221)( 45,222)( 46,241)( 47,242)( 48,243)
( 49,238)( 50,239)( 51,240)( 52,235)( 53,236)( 54,237)( 55,199)( 56,200)
( 57,201)( 58,205)( 59,206)( 60,207)( 61,202)( 62,203)( 63,204)( 64,190)
( 65,191)( 66,192)( 67,196)( 68,197)( 69,198)( 70,193)( 71,194)( 72,195)
( 73,214)( 74,215)( 75,216)( 76,211)( 77,212)( 78,213)( 79,208)( 80,209)
( 81,210)( 82,253)( 83,254)( 84,255)( 85,259)( 86,260)( 87,261)( 88,256)
( 89,257)( 90,258)( 91,244)( 92,245)( 93,246)( 94,250)( 95,251)( 96,252)
( 97,247)( 98,248)( 99,249)(100,268)(101,269)(102,270)(103,265)(104,266)
(105,267)(106,262)(107,263)(108,264)(109,307)(110,308)(111,309)(112,313)
(113,314)(114,315)(115,310)(116,311)(117,312)(118,298)(119,299)(120,300)
(121,304)(122,305)(123,306)(124,301)(125,302)(126,303)(127,322)(128,323)
(129,324)(130,319)(131,320)(132,321)(133,316)(134,317)(135,318)(136,280)
(137,281)(138,282)(139,286)(140,287)(141,288)(142,283)(143,284)(144,285)
(145,271)(146,272)(147,273)(148,277)(149,278)(150,279)(151,274)(152,275)
(153,276)(154,295)(155,296)(156,297)(157,292)(158,293)(159,294)(160,289)
(161,290)(162,291);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope