Polytope of Type {12,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1296d
if this polytope has a name.
Group : SmallGroup(1296,867)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 36, 324, 54
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   2-fold quotients : {6,18}*648h
   3-fold quotients : {12,6}*432a
   4-fold quotients : {3,18}*324
   6-fold quotients : {6,6}*216c
   9-fold quotients : {12,6}*144b
   12-fold quotients : {3,6}*108
   18-fold quotients : {6,6}*72c
   27-fold quotients : {12,2}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*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,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 57)( 29, 56)( 30, 55)( 31, 59)
( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)( 39, 73)
( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)( 47, 65)
( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)(113,139)
(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)(121,158)
(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)(129,145)
(130,149)(131,148)(132,150)(133,151)(134,153)(135,152)(163,244)(164,246)
(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)(172,262)
(173,264)(174,263)(175,267)(176,266)(177,265)(178,269)(179,268)(180,270)
(181,253)(182,255)(183,254)(184,258)(185,257)(186,256)(187,260)(188,259)
(189,261)(190,300)(191,299)(192,298)(193,302)(194,301)(195,303)(196,304)
(197,306)(198,305)(199,318)(200,317)(201,316)(202,320)(203,319)(204,321)
(205,322)(206,324)(207,323)(208,309)(209,308)(210,307)(211,311)(212,310)
(213,312)(214,313)(215,315)(216,314)(217,273)(218,272)(219,271)(220,275)
(221,274)(222,276)(223,277)(224,279)(225,278)(226,291)(227,290)(228,289)
(229,293)(230,292)(231,294)(232,295)(233,297)(234,296)(235,282)(236,281)
(237,280)(238,284)(239,283)(240,285)(241,286)(242,288)(243,287);;
s1 := (  1,229)(  2,231)(  3,230)(  4,232)(  5,234)(  6,233)(  7,226)(  8,228)
(  9,227)( 10,223)( 11,225)( 12,224)( 13,217)( 14,219)( 15,218)( 16,220)
( 17,222)( 18,221)( 19,237)( 20,236)( 21,235)( 22,240)( 23,239)( 24,238)
( 25,243)( 26,242)( 27,241)( 28,202)( 29,204)( 30,203)( 31,205)( 32,207)
( 33,206)( 34,199)( 35,201)( 36,200)( 37,196)( 38,198)( 39,197)( 40,190)
( 41,192)( 42,191)( 43,193)( 44,195)( 45,194)( 46,210)( 47,209)( 48,208)
( 49,213)( 50,212)( 51,211)( 52,216)( 53,215)( 54,214)( 55,175)( 56,177)
( 57,176)( 58,178)( 59,180)( 60,179)( 61,172)( 62,174)( 63,173)( 64,169)
( 65,171)( 66,170)( 67,163)( 68,165)( 69,164)( 70,166)( 71,168)( 72,167)
( 73,183)( 74,182)( 75,181)( 76,186)( 77,185)( 78,184)( 79,189)( 80,188)
( 81,187)( 82,310)( 83,312)( 84,311)( 85,313)( 86,315)( 87,314)( 88,307)
( 89,309)( 90,308)( 91,304)( 92,306)( 93,305)( 94,298)( 95,300)( 96,299)
( 97,301)( 98,303)( 99,302)(100,318)(101,317)(102,316)(103,321)(104,320)
(105,319)(106,324)(107,323)(108,322)(109,283)(110,285)(111,284)(112,286)
(113,288)(114,287)(115,280)(116,282)(117,281)(118,277)(119,279)(120,278)
(121,271)(122,273)(123,272)(124,274)(125,276)(126,275)(127,291)(128,290)
(129,289)(130,294)(131,293)(132,292)(133,297)(134,296)(135,295)(136,256)
(137,258)(138,257)(139,259)(140,261)(141,260)(142,253)(143,255)(144,254)
(145,250)(146,252)(147,251)(148,244)(149,246)(150,245)(151,247)(152,249)
(153,248)(154,264)(155,263)(156,262)(157,267)(158,266)(159,265)(160,270)
(161,269)(162,268);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 57)( 29, 56)( 30, 55)( 31, 63)
( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)
( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)( 47, 74)
( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,138)(110,137)(111,136)(112,144)(113,143)
(114,142)(115,141)(116,140)(117,139)(118,147)(119,146)(120,145)(121,153)
(122,152)(123,151)(124,150)(125,149)(126,148)(127,156)(128,155)(129,154)
(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,219)(191,218)(192,217)(193,225)(194,224)(195,223)
(196,222)(197,221)(198,220)(199,228)(200,227)(201,226)(202,234)(203,233)
(204,232)(205,231)(206,230)(207,229)(208,237)(209,236)(210,235)(211,243)
(212,242)(213,241)(214,240)(215,239)(216,238)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,300)(272,299)(273,298)(274,306)(275,305)(276,304)(277,303)
(278,302)(279,301)(280,309)(281,308)(282,307)(283,315)(284,314)(285,313)
(286,312)(287,311)(288,310)(289,318)(290,317)(291,316)(292,324)(293,323)
(294,322)(295,321)(296,320)(297,319);;
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*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope