Polytope of Type {4,9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,18}*1296
if this polytope has a name.
Group : SmallGroup(1296,1782)
Rank : 4
Schlafli Type : {4,9,18}
Number of vertices, edges, etc : 4, 18, 81, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-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) :
   3-fold quotients : {4,9,6}*432
   9-fold quotients : {4,9,2}*144, {4,3,6}*144
   27-fold quotients : {4,3,2}*48
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, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324);;
s1 := (  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)( 37, 97)( 38, 98)( 39,100)( 40, 99)( 41,105)( 42,106)( 43,108)
( 44,107)( 45,101)( 46,102)( 47,104)( 48,103)( 49, 85)( 50, 86)( 51, 88)
( 52, 87)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)
( 60, 91)( 61, 73)( 62, 74)( 63, 76)( 64, 75)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 77)( 70, 78)( 71, 80)( 72, 79)(109,225)(110,226)(111,228)
(112,227)(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)(119,220)
(120,219)(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)
(128,247)(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)
(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)
(144,231)(145,321)(146,322)(147,324)(148,323)(149,317)(150,318)(151,320)
(152,319)(153,313)(154,314)(155,316)(156,315)(157,309)(158,310)(159,312)
(160,311)(161,305)(162,306)(163,308)(164,307)(165,301)(166,302)(167,304)
(168,303)(169,297)(170,298)(171,300)(172,299)(173,293)(174,294)(175,296)
(176,295)(177,289)(178,290)(179,292)(180,291)(181,285)(182,286)(183,288)
(184,287)(185,281)(186,282)(187,284)(188,283)(189,277)(190,278)(191,280)
(192,279)(193,273)(194,274)(195,276)(196,275)(197,269)(198,270)(199,272)
(200,271)(201,265)(202,266)(203,268)(204,267)(205,261)(206,262)(207,264)
(208,263)(209,257)(210,258)(211,260)(212,259)(213,253)(214,254)(215,256)
(216,255);;
s2 := (  1,145)(  2,148)(  3,147)(  4,146)(  5,153)(  6,156)(  7,155)(  8,154)
(  9,149)( 10,152)( 11,151)( 12,150)( 13,169)( 14,172)( 15,171)( 16,170)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,173)( 22,176)( 23,175)( 24,174)
( 25,157)( 26,160)( 27,159)( 28,158)( 29,165)( 30,168)( 31,167)( 32,166)
( 33,161)( 34,164)( 35,163)( 36,162)( 37,109)( 38,112)( 39,111)( 40,110)
( 41,117)( 42,120)( 43,119)( 44,118)( 45,113)( 46,116)( 47,115)( 48,114)
( 49,133)( 50,136)( 51,135)( 52,134)( 53,141)( 54,144)( 55,143)( 56,142)
( 57,137)( 58,140)( 59,139)( 60,138)( 61,121)( 62,124)( 63,123)( 64,122)
( 65,129)( 66,132)( 67,131)( 68,130)( 69,125)( 70,128)( 71,127)( 72,126)
( 73,205)( 74,208)( 75,207)( 76,206)( 77,213)( 78,216)( 79,215)( 80,214)
( 81,209)( 82,212)( 83,211)( 84,210)( 85,193)( 86,196)( 87,195)( 88,194)
( 89,201)( 90,204)( 91,203)( 92,202)( 93,197)( 94,200)( 95,199)( 96,198)
( 97,181)( 98,184)( 99,183)(100,182)(101,189)(102,192)(103,191)(104,190)
(105,185)(106,188)(107,187)(108,186)(217,261)(218,264)(219,263)(220,262)
(221,257)(222,260)(223,259)(224,258)(225,253)(226,256)(227,255)(228,254)
(229,285)(230,288)(231,287)(232,286)(233,281)(234,284)(235,283)(236,282)
(237,277)(238,280)(239,279)(240,278)(241,273)(242,276)(243,275)(244,274)
(245,269)(246,272)(247,271)(248,270)(249,265)(250,268)(251,267)(252,266)
(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)
(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)
(306,308);;
s3 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,106)( 47,107)( 48,108)
( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 89)( 54, 90)( 55, 91)( 56, 92)
( 57, 93)( 58, 94)( 59, 95)( 60, 96)( 61, 73)( 62, 74)( 63, 75)( 64, 76)
( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 81)( 70, 82)( 71, 83)( 72, 84)
(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)
(129,141)(130,142)(131,143)(132,144)(145,205)(146,206)(147,207)(148,208)
(149,209)(150,210)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)
(157,193)(158,194)(159,195)(160,196)(161,197)(162,198)(163,199)(164,200)
(165,201)(166,202)(167,203)(168,204)(169,181)(170,182)(171,183)(172,184)
(173,185)(174,186)(175,187)(176,188)(177,189)(178,190)(179,191)(180,192)
(229,241)(230,242)(231,243)(232,244)(233,245)(234,246)(235,247)(236,248)
(237,249)(238,250)(239,251)(240,252)(253,313)(254,314)(255,315)(256,316)
(257,317)(258,318)(259,319)(260,320)(261,321)(262,322)(263,323)(264,324)
(265,301)(266,302)(267,303)(268,304)(269,305)(270,306)(271,307)(272,308)
(273,309)(274,310)(275,311)(276,312)(277,289)(278,290)(279,291)(280,292)
(281,293)(282,294)(283,295)(284,296)(285,297)(286,298)(287,299)(288,300);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*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)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324);
s1 := 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)( 37, 97)( 38, 98)( 39,100)( 40, 99)( 41,105)( 42,106)
( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)( 48,103)( 49, 85)( 50, 86)
( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)
( 59, 92)( 60, 91)( 61, 73)( 62, 74)( 63, 76)( 64, 75)( 65, 81)( 66, 82)
( 67, 84)( 68, 83)( 69, 77)( 70, 78)( 71, 80)( 72, 79)(109,225)(110,226)
(111,228)(112,227)(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)
(119,220)(120,219)(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)
(127,248)(128,247)(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)
(135,240)(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)
(143,232)(144,231)(145,321)(146,322)(147,324)(148,323)(149,317)(150,318)
(151,320)(152,319)(153,313)(154,314)(155,316)(156,315)(157,309)(158,310)
(159,312)(160,311)(161,305)(162,306)(163,308)(164,307)(165,301)(166,302)
(167,304)(168,303)(169,297)(170,298)(171,300)(172,299)(173,293)(174,294)
(175,296)(176,295)(177,289)(178,290)(179,292)(180,291)(181,285)(182,286)
(183,288)(184,287)(185,281)(186,282)(187,284)(188,283)(189,277)(190,278)
(191,280)(192,279)(193,273)(194,274)(195,276)(196,275)(197,269)(198,270)
(199,272)(200,271)(201,265)(202,266)(203,268)(204,267)(205,261)(206,262)
(207,264)(208,263)(209,257)(210,258)(211,260)(212,259)(213,253)(214,254)
(215,256)(216,255);
s2 := Sym(324)!(  1,145)(  2,148)(  3,147)(  4,146)(  5,153)(  6,156)(  7,155)
(  8,154)(  9,149)( 10,152)( 11,151)( 12,150)( 13,169)( 14,172)( 15,171)
( 16,170)( 17,177)( 18,180)( 19,179)( 20,178)( 21,173)( 22,176)( 23,175)
( 24,174)( 25,157)( 26,160)( 27,159)( 28,158)( 29,165)( 30,168)( 31,167)
( 32,166)( 33,161)( 34,164)( 35,163)( 36,162)( 37,109)( 38,112)( 39,111)
( 40,110)( 41,117)( 42,120)( 43,119)( 44,118)( 45,113)( 46,116)( 47,115)
( 48,114)( 49,133)( 50,136)( 51,135)( 52,134)( 53,141)( 54,144)( 55,143)
( 56,142)( 57,137)( 58,140)( 59,139)( 60,138)( 61,121)( 62,124)( 63,123)
( 64,122)( 65,129)( 66,132)( 67,131)( 68,130)( 69,125)( 70,128)( 71,127)
( 72,126)( 73,205)( 74,208)( 75,207)( 76,206)( 77,213)( 78,216)( 79,215)
( 80,214)( 81,209)( 82,212)( 83,211)( 84,210)( 85,193)( 86,196)( 87,195)
( 88,194)( 89,201)( 90,204)( 91,203)( 92,202)( 93,197)( 94,200)( 95,199)
( 96,198)( 97,181)( 98,184)( 99,183)(100,182)(101,189)(102,192)(103,191)
(104,190)(105,185)(106,188)(107,187)(108,186)(217,261)(218,264)(219,263)
(220,262)(221,257)(222,260)(223,259)(224,258)(225,253)(226,256)(227,255)
(228,254)(229,285)(230,288)(231,287)(232,286)(233,281)(234,284)(235,283)
(236,282)(237,277)(238,280)(239,279)(240,278)(241,273)(242,276)(243,275)
(244,274)(245,269)(246,272)(247,271)(248,270)(249,265)(250,268)(251,267)
(252,266)(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)
(296,318)(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)
(304,310)(306,308);
s3 := Sym(324)!( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,106)( 47,107)
( 48,108)( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 89)( 54, 90)( 55, 91)
( 56, 92)( 57, 93)( 58, 94)( 59, 95)( 60, 96)( 61, 73)( 62, 74)( 63, 75)
( 64, 76)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 81)( 70, 82)( 71, 83)
( 72, 84)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)
(128,140)(129,141)(130,142)(131,143)(132,144)(145,205)(146,206)(147,207)
(148,208)(149,209)(150,210)(151,211)(152,212)(153,213)(154,214)(155,215)
(156,216)(157,193)(158,194)(159,195)(160,196)(161,197)(162,198)(163,199)
(164,200)(165,201)(166,202)(167,203)(168,204)(169,181)(170,182)(171,183)
(172,184)(173,185)(174,186)(175,187)(176,188)(177,189)(178,190)(179,191)
(180,192)(229,241)(230,242)(231,243)(232,244)(233,245)(234,246)(235,247)
(236,248)(237,249)(238,250)(239,251)(240,252)(253,313)(254,314)(255,315)
(256,316)(257,317)(258,318)(259,319)(260,320)(261,321)(262,322)(263,323)
(264,324)(265,301)(266,302)(267,303)(268,304)(269,305)(270,306)(271,307)
(272,308)(273,309)(274,310)(275,311)(276,312)(277,289)(278,290)(279,291)
(280,292)(281,293)(282,294)(283,295)(284,296)(285,297)(286,298)(287,299)
(288,300);
poly := sub<Sym(324)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope