Polytope of Type {36,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,18}*1296c
if this polytope has a name.
Group : SmallGroup(1296,878)
Rank : 3
Schlafli Type : {36,18}
Number of vertices, edges, etc : 36, 324, 18
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
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 : {18,18}*648b
   3-fold quotients : {12,18}*432b
   4-fold quotients : {18,9}*324
   6-fold quotients : {6,18}*216b
   9-fold quotients : {4,18}*144a, {12,6}*144c
   12-fold quotients : {6,9}*108
   18-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {4,6}*48a
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)
( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)( 42, 69)
( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)
( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)(111,144)
(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)(119,152)
(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(127,160)
(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)
(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)
(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)
(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)
(187,265)(188,266)(189,267)(190,304)(191,305)(192,306)(193,301)(194,302)
(195,303)(196,298)(197,299)(198,300)(199,313)(200,314)(201,315)(202,310)
(203,311)(204,312)(205,307)(206,308)(207,309)(208,322)(209,323)(210,324)
(211,319)(212,320)(213,321)(214,316)(215,317)(216,318)(217,277)(218,278)
(219,279)(220,274)(221,275)(222,276)(223,271)(224,272)(225,273)(226,286)
(227,287)(228,288)(229,283)(230,284)(231,285)(232,280)(233,281)(234,282)
(235,295)(236,296)(237,297)(238,292)(239,293)(240,294)(241,289)(242,290)
(243,291);;
s1 := (  1,190)(  2,192)(  3,191)(  4,196)(  5,198)(  6,197)(  7,193)(  8,195)
(  9,194)( 10,210)( 11,209)( 12,208)( 13,216)( 14,215)( 15,214)( 16,213)
( 17,212)( 18,211)( 19,201)( 20,200)( 21,199)( 22,207)( 23,206)( 24,205)
( 25,204)( 26,203)( 27,202)( 28,163)( 29,165)( 30,164)( 31,169)( 32,171)
( 33,170)( 34,166)( 35,168)( 36,167)( 37,183)( 38,182)( 39,181)( 40,189)
( 41,188)( 42,187)( 43,186)( 44,185)( 45,184)( 46,174)( 47,173)( 48,172)
( 49,180)( 50,179)( 51,178)( 52,177)( 53,176)( 54,175)( 55,223)( 56,225)
( 57,224)( 58,220)( 59,222)( 60,221)( 61,217)( 62,219)( 63,218)( 64,243)
( 65,242)( 66,241)( 67,240)( 68,239)( 69,238)( 70,237)( 71,236)( 72,235)
( 73,234)( 74,233)( 75,232)( 76,231)( 77,230)( 78,229)( 79,228)( 80,227)
( 81,226)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)( 88,274)
( 89,276)( 90,275)( 91,291)( 92,290)( 93,289)( 94,297)( 95,296)( 96,295)
( 97,294)( 98,293)( 99,292)(100,282)(101,281)(102,280)(103,288)(104,287)
(105,286)(106,285)(107,284)(108,283)(109,244)(110,246)(111,245)(112,250)
(113,252)(114,251)(115,247)(116,249)(117,248)(118,264)(119,263)(120,262)
(121,270)(122,269)(123,268)(124,267)(125,266)(126,265)(127,255)(128,254)
(129,253)(130,261)(131,260)(132,259)(133,258)(134,257)(135,256)(136,304)
(137,306)(138,305)(139,301)(140,303)(141,302)(142,298)(143,300)(144,299)
(145,324)(146,323)(147,322)(148,321)(149,320)(150,319)(151,318)(152,317)
(153,316)(154,315)(155,314)(156,313)(157,312)(158,311)(159,310)(160,309)
(161,308)(162,307);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 70)( 29, 72)( 30, 71)
( 31, 67)( 32, 69)( 33, 68)( 34, 64)( 35, 66)( 36, 65)( 37, 61)( 38, 63)
( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 81)
( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,151)(110,153)(111,152)
(112,148)(113,150)(114,149)(115,145)(116,147)(117,146)(118,142)(119,144)
(120,143)(121,139)(122,141)(123,140)(124,136)(125,138)(126,137)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(163,172)(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)
(171,176)(181,183)(184,189)(185,188)(186,187)(190,232)(191,234)(192,233)
(193,229)(194,231)(195,230)(196,226)(197,228)(198,227)(199,223)(200,225)
(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,243)
(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)
(244,253)(245,255)(246,254)(247,259)(248,261)(249,260)(250,256)(251,258)
(252,257)(262,264)(265,270)(266,269)(267,268)(271,313)(272,315)(273,314)
(274,310)(275,312)(276,311)(277,307)(278,309)(279,308)(280,304)(281,306)
(282,305)(283,301)(284,303)(285,302)(286,298)(287,300)(288,299)(289,324)
(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)(297,316);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope