Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1296k
if this polytope has a name.
Group : SmallGroup(1296,1789)
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 : {18,12}*432c, {6,12}*432d
   4-fold quotients : {18,6}*324a
   9-fold quotients : {18,4}*144c, {6,12}*144d
   12-fold quotients : {6,6}*108
   18-fold quotients : {9,4}*72
   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)(109,241)(110,242)(111,244)(112,243)(113,249)
(114,250)(115,252)(116,251)(117,245)(118,246)(119,248)(120,247)(121,229)
(122,230)(123,232)(124,231)(125,237)(126,238)(127,240)(128,239)(129,233)
(130,234)(131,236)(132,235)(133,217)(134,218)(135,220)(136,219)(137,225)
(138,226)(139,228)(140,227)(141,221)(142,222)(143,224)(144,223)(145,277)
(146,278)(147,280)(148,279)(149,285)(150,286)(151,288)(152,287)(153,281)
(154,282)(155,284)(156,283)(157,265)(158,266)(159,268)(160,267)(161,273)
(162,274)(163,276)(164,275)(165,269)(166,270)(167,272)(168,271)(169,253)
(170,254)(171,256)(172,255)(173,261)(174,262)(175,264)(176,263)(177,257)
(178,258)(179,260)(180,259)(181,313)(182,314)(183,316)(184,315)(185,321)
(186,322)(187,324)(188,323)(189,317)(190,318)(191,320)(192,319)(193,301)
(194,302)(195,304)(196,303)(197,309)(198,310)(199,312)(200,311)(201,305)
(202,306)(203,308)(204,307)(205,289)(206,290)(207,292)(208,291)(209,297)
(210,298)(211,300)(212,299)(213,293)(214,294)(215,296)(216,295);;
s1 := (  1,109)(  2,112)(  3,111)(  4,110)(  5,113)(  6,116)(  7,115)(  8,114)
(  9,117)( 10,120)( 11,119)( 12,118)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,137)( 18,140)( 19,139)( 20,138)( 21,141)( 22,144)( 23,143)( 24,142)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,125)( 30,128)( 31,127)( 32,126)
( 33,129)( 34,132)( 35,131)( 36,130)( 37,189)( 38,192)( 39,191)( 40,190)
( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,213)( 50,216)( 51,215)( 52,214)( 53,205)( 54,208)( 55,207)( 56,206)
( 57,209)( 58,212)( 59,211)( 60,210)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,193)( 66,196)( 67,195)( 68,194)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,149)( 74,152)( 75,151)( 76,150)( 77,153)( 78,156)( 79,155)( 80,154)
( 81,145)( 82,148)( 83,147)( 84,146)( 85,173)( 86,176)( 87,175)( 88,174)
( 89,177)( 90,180)( 91,179)( 92,178)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,161)( 98,164)( 99,163)(100,162)(101,165)(102,168)(103,167)(104,166)
(105,157)(106,160)(107,159)(108,158)(217,241)(218,244)(219,243)(220,242)
(221,245)(222,248)(223,247)(224,246)(225,249)(226,252)(227,251)(228,250)
(230,232)(234,236)(238,240)(253,321)(254,324)(255,323)(256,322)(257,313)
(258,316)(259,315)(260,314)(261,317)(262,320)(263,319)(264,318)(265,309)
(266,312)(267,311)(268,310)(269,301)(270,304)(271,303)(272,302)(273,305)
(274,308)(275,307)(276,306)(277,297)(278,300)(279,299)(280,298)(281,289)
(282,292)(283,291)(284,290)(285,293)(286,296)(287,295)(288,294);;
s2 := (  1, 38)(  2, 37)(  3, 40)(  4, 39)(  5, 46)(  6, 45)(  7, 48)(  8, 47)
(  9, 42)( 10, 41)( 11, 44)( 12, 43)( 13, 50)( 14, 49)( 15, 52)( 16, 51)
( 17, 58)( 18, 57)( 19, 60)( 20, 59)( 21, 54)( 22, 53)( 23, 56)( 24, 55)
( 25, 62)( 26, 61)( 27, 64)( 28, 63)( 29, 70)( 30, 69)( 31, 72)( 32, 71)
( 33, 66)( 34, 65)( 35, 68)( 36, 67)( 73, 74)( 75, 76)( 77, 82)( 78, 81)
( 79, 84)( 80, 83)( 85, 86)( 87, 88)( 89, 94)( 90, 93)( 91, 96)( 92, 95)
( 97, 98)( 99,100)(101,106)(102,105)(103,108)(104,107)(109,146)(110,145)
(111,148)(112,147)(113,154)(114,153)(115,156)(116,155)(117,150)(118,149)
(119,152)(120,151)(121,158)(122,157)(123,160)(124,159)(125,166)(126,165)
(127,168)(128,167)(129,162)(130,161)(131,164)(132,163)(133,170)(134,169)
(135,172)(136,171)(137,178)(138,177)(139,180)(140,179)(141,174)(142,173)
(143,176)(144,175)(181,182)(183,184)(185,190)(186,189)(187,192)(188,191)
(193,194)(195,196)(197,202)(198,201)(199,204)(200,203)(205,206)(207,208)
(209,214)(210,213)(211,216)(212,215)(217,254)(218,253)(219,256)(220,255)
(221,262)(222,261)(223,264)(224,263)(225,258)(226,257)(227,260)(228,259)
(229,266)(230,265)(231,268)(232,267)(233,274)(234,273)(235,276)(236,275)
(237,270)(238,269)(239,272)(240,271)(241,278)(242,277)(243,280)(244,279)
(245,286)(246,285)(247,288)(248,287)(249,282)(250,281)(251,284)(252,283)
(289,290)(291,292)(293,298)(294,297)(295,300)(296,299)(301,302)(303,304)
(305,310)(306,309)(307,312)(308,311)(313,314)(315,316)(317,322)(318,321)
(319,324)(320,323);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 ];;
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)(109,241)(110,242)(111,244)(112,243)
(113,249)(114,250)(115,252)(116,251)(117,245)(118,246)(119,248)(120,247)
(121,229)(122,230)(123,232)(124,231)(125,237)(126,238)(127,240)(128,239)
(129,233)(130,234)(131,236)(132,235)(133,217)(134,218)(135,220)(136,219)
(137,225)(138,226)(139,228)(140,227)(141,221)(142,222)(143,224)(144,223)
(145,277)(146,278)(147,280)(148,279)(149,285)(150,286)(151,288)(152,287)
(153,281)(154,282)(155,284)(156,283)(157,265)(158,266)(159,268)(160,267)
(161,273)(162,274)(163,276)(164,275)(165,269)(166,270)(167,272)(168,271)
(169,253)(170,254)(171,256)(172,255)(173,261)(174,262)(175,264)(176,263)
(177,257)(178,258)(179,260)(180,259)(181,313)(182,314)(183,316)(184,315)
(185,321)(186,322)(187,324)(188,323)(189,317)(190,318)(191,320)(192,319)
(193,301)(194,302)(195,304)(196,303)(197,309)(198,310)(199,312)(200,311)
(201,305)(202,306)(203,308)(204,307)(205,289)(206,290)(207,292)(208,291)
(209,297)(210,298)(211,300)(212,299)(213,293)(214,294)(215,296)(216,295);
s1 := Sym(324)!(  1,109)(  2,112)(  3,111)(  4,110)(  5,113)(  6,116)(  7,115)
(  8,114)(  9,117)( 10,120)( 11,119)( 12,118)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,137)( 18,140)( 19,139)( 20,138)( 21,141)( 22,144)( 23,143)
( 24,142)( 25,121)( 26,124)( 27,123)( 28,122)( 29,125)( 30,128)( 31,127)
( 32,126)( 33,129)( 34,132)( 35,131)( 36,130)( 37,189)( 38,192)( 39,191)
( 40,190)( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)
( 48,186)( 49,213)( 50,216)( 51,215)( 52,214)( 53,205)( 54,208)( 55,207)
( 56,206)( 57,209)( 58,212)( 59,211)( 60,210)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,193)( 66,196)( 67,195)( 68,194)( 69,197)( 70,200)( 71,199)
( 72,198)( 73,149)( 74,152)( 75,151)( 76,150)( 77,153)( 78,156)( 79,155)
( 80,154)( 81,145)( 82,148)( 83,147)( 84,146)( 85,173)( 86,176)( 87,175)
( 88,174)( 89,177)( 90,180)( 91,179)( 92,178)( 93,169)( 94,172)( 95,171)
( 96,170)( 97,161)( 98,164)( 99,163)(100,162)(101,165)(102,168)(103,167)
(104,166)(105,157)(106,160)(107,159)(108,158)(217,241)(218,244)(219,243)
(220,242)(221,245)(222,248)(223,247)(224,246)(225,249)(226,252)(227,251)
(228,250)(230,232)(234,236)(238,240)(253,321)(254,324)(255,323)(256,322)
(257,313)(258,316)(259,315)(260,314)(261,317)(262,320)(263,319)(264,318)
(265,309)(266,312)(267,311)(268,310)(269,301)(270,304)(271,303)(272,302)
(273,305)(274,308)(275,307)(276,306)(277,297)(278,300)(279,299)(280,298)
(281,289)(282,292)(283,291)(284,290)(285,293)(286,296)(287,295)(288,294);
s2 := Sym(324)!(  1, 38)(  2, 37)(  3, 40)(  4, 39)(  5, 46)(  6, 45)(  7, 48)
(  8, 47)(  9, 42)( 10, 41)( 11, 44)( 12, 43)( 13, 50)( 14, 49)( 15, 52)
( 16, 51)( 17, 58)( 18, 57)( 19, 60)( 20, 59)( 21, 54)( 22, 53)( 23, 56)
( 24, 55)( 25, 62)( 26, 61)( 27, 64)( 28, 63)( 29, 70)( 30, 69)( 31, 72)
( 32, 71)( 33, 66)( 34, 65)( 35, 68)( 36, 67)( 73, 74)( 75, 76)( 77, 82)
( 78, 81)( 79, 84)( 80, 83)( 85, 86)( 87, 88)( 89, 94)( 90, 93)( 91, 96)
( 92, 95)( 97, 98)( 99,100)(101,106)(102,105)(103,108)(104,107)(109,146)
(110,145)(111,148)(112,147)(113,154)(114,153)(115,156)(116,155)(117,150)
(118,149)(119,152)(120,151)(121,158)(122,157)(123,160)(124,159)(125,166)
(126,165)(127,168)(128,167)(129,162)(130,161)(131,164)(132,163)(133,170)
(134,169)(135,172)(136,171)(137,178)(138,177)(139,180)(140,179)(141,174)
(142,173)(143,176)(144,175)(181,182)(183,184)(185,190)(186,189)(187,192)
(188,191)(193,194)(195,196)(197,202)(198,201)(199,204)(200,203)(205,206)
(207,208)(209,214)(210,213)(211,216)(212,215)(217,254)(218,253)(219,256)
(220,255)(221,262)(222,261)(223,264)(224,263)(225,258)(226,257)(227,260)
(228,259)(229,266)(230,265)(231,268)(232,267)(233,274)(234,273)(235,276)
(236,275)(237,270)(238,269)(239,272)(240,271)(241,278)(242,277)(243,280)
(244,279)(245,286)(246,285)(247,288)(248,287)(249,282)(250,281)(251,284)
(252,283)(289,290)(291,292)(293,298)(294,297)(295,300)(296,299)(301,302)
(303,304)(305,310)(306,309)(307,312)(308,311)(313,314)(315,316)(317,322)
(318,321)(319,324)(320,323);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 >;

References : None.
to this polytope