Polytope of Type {6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1296c
if this polytope has a name.
Group : SmallGroup(1296,839)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s₀s₁s₂ : 12
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}*648d
   3-fold quotients : {6,12}*432a
   4-fold quotients : {6,9}*324b
   6-fold quotients : {6,6}*216a
   9-fold quotients : {6,12}*144b
   12-fold quotients : {6,3}*108
   18-fold quotients : {6,6}*72b
   27-fold quotients : {2,12}*48
   36-fold quotients : {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)(  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);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)
(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)
( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)
( 25, 38)( 26, 37)( 27, 39)( 56, 57)( 58, 60)( 61, 62)( 64, 80)( 65, 79)
( 66, 81)( 67, 73)( 68, 75)( 69, 74)( 70, 78)( 71, 77)( 72, 76)( 82,109)
( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)
( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)
( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)
(107,118)(108,120)(137,138)(139,141)(142,143)(145,161)(146,160)(147,162)
(148,154)(149,156)(150,155)(151,159)(152,158)(153,157)(163,271)(164,273)
(165,272)(166,276)(167,275)(168,274)(169,278)(170,277)(171,279)(172,296)
(173,295)(174,297)(175,289)(176,291)(177,290)(178,294)(179,293)(180,292)
(181,283)(182,285)(183,284)(184,288)(185,287)(186,286)(187,281)(188,280)
(189,282)(190,244)(191,246)(192,245)(193,249)(194,248)(195,247)(196,251)
(197,250)(198,252)(199,269)(200,268)(201,270)(202,262)(203,264)(204,263)
(205,267)(206,266)(207,265)(208,256)(209,258)(210,257)(211,261)(212,260)
(213,259)(214,254)(215,253)(216,255)(217,298)(218,300)(219,299)(220,303)
(221,302)(222,301)(223,305)(224,304)(225,306)(226,323)(227,322)(228,324)
(229,316)(230,318)(231,317)(232,321)(233,320)(234,319)(235,310)(236,312)
(237,311)(238,315)(239,314)(240,313)(241,308)(242,307)(243,309);;
s2 := (  1,172)(  2,174)(  3,173)(  4,175)(  5,177)(  6,176)(  7,178)(  8,180)
(  9,179)( 10,163)( 11,165)( 12,164)( 13,166)( 14,168)( 15,167)( 16,169)
( 17,171)( 18,170)( 19,181)( 20,183)( 21,182)( 22,184)( 23,186)( 24,185)
( 25,187)( 26,189)( 27,188)( 28,226)( 29,228)( 30,227)( 31,229)( 32,231)
( 33,230)( 34,232)( 35,234)( 36,233)( 37,217)( 38,219)( 39,218)( 40,220)
( 41,222)( 42,221)( 43,223)( 44,225)( 45,224)( 46,235)( 47,237)( 48,236)
( 49,238)( 50,240)( 51,239)( 52,241)( 53,243)( 54,242)( 55,199)( 56,201)
( 57,200)( 58,202)( 59,204)( 60,203)( 61,205)( 62,207)( 63,206)( 64,190)
( 65,192)( 66,191)( 67,193)( 68,195)( 69,194)( 70,196)( 71,198)( 72,197)
( 73,208)( 74,210)( 75,209)( 76,211)( 77,213)( 78,212)( 79,214)( 80,216)
( 81,215)( 82,253)( 83,255)( 84,254)( 85,256)( 86,258)( 87,257)( 88,259)
( 89,261)( 90,260)( 91,244)( 92,246)( 93,245)( 94,247)( 95,249)( 96,248)
( 97,250)( 98,252)( 99,251)(100,262)(101,264)(102,263)(103,265)(104,267)
(105,266)(106,268)(107,270)(108,269)(109,307)(110,309)(111,308)(112,310)
(113,312)(114,311)(115,313)(116,315)(117,314)(118,298)(119,300)(120,299)
(121,301)(122,303)(123,302)(124,304)(125,306)(126,305)(127,316)(128,318)
(129,317)(130,319)(131,321)(132,320)(133,322)(134,324)(135,323)(136,280)
(137,282)(138,281)(139,283)(140,285)(141,284)(142,286)(143,288)(144,287)
(145,271)(146,273)(147,272)(148,274)(149,276)(150,275)(151,277)(152,279)
(153,278)(154,289)(155,291)(156,290)(157,292)(158,294)(159,293)(160,295)
(161,297)(162,296);;
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, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope