Polytope of Type {18,36}

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

Permutation Representation (Magma) :

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

References : None.
to this polytope