Polytope of Type {4,18,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1789)
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)(109,241)(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);;
s2 := ( 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);;
s3 := ( 1, 37)( 2, 38)( 3, 39)( 4, 40)( 5, 45)( 6, 46)( 7, 47)( 8, 48)
( 9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 49)( 14, 50)( 15, 51)( 16, 52)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 53)( 22, 54)( 23, 55)( 24, 56)
( 25, 61)( 26, 62)( 27, 63)( 28, 64)( 29, 69)( 30, 70)( 31, 71)( 32, 72)
( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 77, 81)( 78, 82)( 79, 83)( 80, 84)
( 89, 93)( 90, 94)( 91, 95)( 92, 96)(101,105)(102,106)(103,107)(104,108)
(109,145)(110,146)(111,147)(112,148)(113,153)(114,154)(115,155)(116,156)
(117,149)(118,150)(119,151)(120,152)(121,157)(122,158)(123,159)(124,160)
(125,165)(126,166)(127,167)(128,168)(129,161)(130,162)(131,163)(132,164)
(133,169)(134,170)(135,171)(136,172)(137,177)(138,178)(139,179)(140,180)
(141,173)(142,174)(143,175)(144,176)(185,189)(186,190)(187,191)(188,192)
(197,201)(198,202)(199,203)(200,204)(209,213)(210,214)(211,215)(212,216)
(217,253)(218,254)(219,255)(220,256)(221,261)(222,262)(223,263)(224,264)
(225,257)(226,258)(227,259)(228,260)(229,265)(230,266)(231,267)(232,268)
(233,273)(234,274)(235,275)(236,276)(237,269)(238,270)(239,271)(240,272)
(241,277)(242,278)(243,279)(244,280)(245,285)(246,286)(247,287)(248,288)
(249,281)(250,282)(251,283)(252,284)(293,297)(294,298)(295,299)(296,300)
(305,309)(306,310)(307,311)(308,312)(317,321)(318,322)(319,323)(320,324);;
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*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*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)( 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);
s2 := 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);
s3 := Sym(324)!( 1, 37)( 2, 38)( 3, 39)( 4, 40)( 5, 45)( 6, 46)( 7, 47)
( 8, 48)( 9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 49)( 14, 50)( 15, 51)
( 16, 52)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 53)( 22, 54)( 23, 55)
( 24, 56)( 25, 61)( 26, 62)( 27, 63)( 28, 64)( 29, 69)( 30, 70)( 31, 71)
( 32, 72)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 77, 81)( 78, 82)( 79, 83)
( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)(101,105)(102,106)(103,107)
(104,108)(109,145)(110,146)(111,147)(112,148)(113,153)(114,154)(115,155)
(116,156)(117,149)(118,150)(119,151)(120,152)(121,157)(122,158)(123,159)
(124,160)(125,165)(126,166)(127,167)(128,168)(129,161)(130,162)(131,163)
(132,164)(133,169)(134,170)(135,171)(136,172)(137,177)(138,178)(139,179)
(140,180)(141,173)(142,174)(143,175)(144,176)(185,189)(186,190)(187,191)
(188,192)(197,201)(198,202)(199,203)(200,204)(209,213)(210,214)(211,215)
(212,216)(217,253)(218,254)(219,255)(220,256)(221,261)(222,262)(223,263)
(224,264)(225,257)(226,258)(227,259)(228,260)(229,265)(230,266)(231,267)
(232,268)(233,273)(234,274)(235,275)(236,276)(237,269)(238,270)(239,271)
(240,272)(241,277)(242,278)(243,279)(244,280)(245,285)(246,286)(247,287)
(248,288)(249,281)(250,282)(251,283)(252,284)(293,297)(294,298)(295,299)
(296,300)(305,309)(306,310)(307,311)(308,312)(317,321)(318,322)(319,323)
(320,324);
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*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 >;
References : None.
to this polytope