Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1296j
if this polytope has a name.
Group : SmallGroup(1296,1787)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 54, 324, 36
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
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 : {6,12}*432d
   4-fold quotients : {18,6}*324b
   9-fold quotients : {6,12}*144d
   12-fold quotients : {6,6}*108
   27-fold quotients : {6,4}*48b
   54-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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