Polytope of Type {36,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,18}*1296d
if this polytope has a name.
Group : SmallGroup(1296,1782)
Rank : 3
Schlafli Type : {36,18}
Number of vertices, edges, etc : 36, 324, 18
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
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) :
   3-fold quotients : {36,6}*432c, {12,18}*432c
   9-fold quotients : {4,18}*144c, {12,6}*144d
   18-fold quotients : {4,9}*72
   27-fold quotients : {4,6}*48b
   54-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope