Polytope of Type {4,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1787)
Rank : 4
Schlafli Type : {4,18,6}
Number of vertices, edges, etc : 4, 54, 81, 9
Order of s₀s₁s₂s₃ : 9
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 : {4,6,6}*432
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)( 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);;
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)( 13,129)( 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);;
s3 := (  1, 37)(  2, 38)(  3, 39)(  4, 40)(  5, 41)(  6, 42)(  7, 43)(  8, 44)
(  9, 45)( 10, 46)( 11, 47)( 12, 48)( 13, 61)( 14, 62)( 15, 63)( 16, 64)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 49)( 26, 50)( 27, 51)( 28, 52)( 29, 53)( 30, 54)( 31, 55)( 32, 56)
( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 85, 97)( 86, 98)( 87, 99)( 88,100)
( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)
(109,145)(110,146)(111,147)(112,148)(113,149)(114,150)(115,151)(116,152)
(117,153)(118,154)(119,155)(120,156)(121,169)(122,170)(123,171)(124,172)
(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)
(133,157)(134,158)(135,159)(136,160)(137,161)(138,162)(139,163)(140,164)
(141,165)(142,166)(143,167)(144,168)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)
(217,253)(218,254)(219,255)(220,256)(221,257)(222,258)(223,259)(224,260)
(225,261)(226,262)(227,263)(228,264)(229,277)(230,278)(231,279)(232,280)
(233,281)(234,282)(235,283)(236,284)(237,285)(238,286)(239,287)(240,288)
(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)(248,272)
(249,273)(250,274)(251,275)(252,276)(301,313)(302,314)(303,315)(304,316)
(305,317)(306,318)(307,319)(308,320)(309,321)(310,322)(311,323)(312,324);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s3*s2*s3*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)( 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);
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)( 13,129)( 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);
s3 := Sym(324)!(  1, 37)(  2, 38)(  3, 39)(  4, 40)(  5, 41)(  6, 42)(  7, 43)
(  8, 44)(  9, 45)( 10, 46)( 11, 47)( 12, 48)( 13, 61)( 14, 62)( 15, 63)
( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)
( 24, 72)( 25, 49)( 26, 50)( 27, 51)( 28, 52)( 29, 53)( 30, 54)( 31, 55)
( 32, 56)( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 85, 97)( 86, 98)( 87, 99)
( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)( 95,107)
( 96,108)(109,145)(110,146)(111,147)(112,148)(113,149)(114,150)(115,151)
(116,152)(117,153)(118,154)(119,155)(120,156)(121,169)(122,170)(123,171)
(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)
(132,180)(133,157)(134,158)(135,159)(136,160)(137,161)(138,162)(139,163)
(140,164)(141,165)(142,166)(143,167)(144,168)(193,205)(194,206)(195,207)
(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)
(204,216)(217,253)(218,254)(219,255)(220,256)(221,257)(222,258)(223,259)
(224,260)(225,261)(226,262)(227,263)(228,264)(229,277)(230,278)(231,279)
(232,280)(233,281)(234,282)(235,283)(236,284)(237,285)(238,286)(239,287)
(240,288)(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)
(248,272)(249,273)(250,274)(251,275)(252,276)(301,313)(302,314)(303,315)
(304,316)(305,317)(306,318)(307,319)(308,320)(309,321)(310,322)(311,323)
(312,324);
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s3*s2*s3*s2 >;

References : None.
to this polytope