Polytope of Type {3,6,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,9,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,1789)
Rank : 5
Schlafli Type : {3,6,9,4}
Number of vertices, edges, etc : 3, 9, 27, 18, 4
Order of s0s1s2s3s4 : 9
Order of s0s1s2s3s4s3s2s1 : 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 : {3,2,9,4}*432, {3,6,3,4}*432
9-fold quotients : {3,2,3,4}*144
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 37, 73)( 38, 74)( 39, 75)( 40, 76)
( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 77)( 46, 78)( 47, 79)( 48, 80)
( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 93)( 54, 94)( 55, 95)( 56, 96)
( 57, 89)( 58, 90)( 59, 91)( 60, 92)( 61, 97)( 62, 98)( 63, 99)( 64,100)
( 65,105)( 66,106)( 67,107)( 68,108)( 69,101)( 70,102)( 71,103)( 72,104)
(113,117)(114,118)(115,119)(116,120)(125,129)(126,130)(127,131)(128,132)
(137,141)(138,142)(139,143)(140,144)(145,181)(146,182)(147,183)(148,184)
(149,189)(150,190)(151,191)(152,192)(153,185)(154,186)(155,187)(156,188)
(157,193)(158,194)(159,195)(160,196)(161,201)(162,202)(163,203)(164,204)
(165,197)(166,198)(167,199)(168,200)(169,205)(170,206)(171,207)(172,208)
(173,213)(174,214)(175,215)(176,216)(177,209)(178,210)(179,211)(180,212)
(221,225)(222,226)(223,227)(224,228)(233,237)(234,238)(235,239)(236,240)
(245,249)(246,250)(247,251)(248,252)(253,289)(254,290)(255,291)(256,292)
(257,297)(258,298)(259,299)(260,300)(261,293)(262,294)(263,295)(264,296)
(265,301)(266,302)(267,303)(268,304)(269,309)(270,310)(271,311)(272,312)
(273,305)(274,306)(275,307)(276,308)(277,313)(278,314)(279,315)(280,316)
(281,321)(282,322)(283,323)(284,324)(285,317)(286,318)(287,319)(288,320);;
s1 := ( 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);;
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)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)( 24,138)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)( 32,130)
( 33,125)( 34,128)( 35,127)( 36,126)( 37,149)( 38,152)( 39,151)( 40,150)
( 41,145)( 42,148)( 43,147)( 44,146)( 45,153)( 46,156)( 47,155)( 48,154)
( 49,173)( 50,176)( 51,175)( 52,174)( 53,169)( 54,172)( 55,171)( 56,170)
( 57,177)( 58,180)( 59,179)( 60,178)( 61,161)( 62,164)( 63,163)( 64,162)
( 65,157)( 66,160)( 67,159)( 68,158)( 69,165)( 70,168)( 71,167)( 72,166)
( 73,189)( 74,192)( 75,191)( 76,190)( 77,185)( 78,188)( 79,187)( 80,186)
( 81,181)( 82,184)( 83,183)( 84,182)( 85,213)( 86,216)( 87,215)( 88,214)
( 89,209)( 90,212)( 91,211)( 92,210)( 93,205)( 94,208)( 95,207)( 96,206)
( 97,201)( 98,204)( 99,203)(100,202)(101,197)(102,200)(103,199)(104,198)
(105,193)(106,196)(107,195)(108,194)(217,241)(218,244)(219,243)(220,242)
(221,249)(222,252)(223,251)(224,250)(225,245)(226,248)(227,247)(228,246)
(230,232)(233,237)(234,240)(235,239)(236,238)(253,281)(254,284)(255,283)
(256,282)(257,277)(258,280)(259,279)(260,278)(261,285)(262,288)(263,287)
(264,286)(265,269)(266,272)(267,271)(268,270)(274,276)(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 := ( 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);;
s4 := ( 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);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s4*s3*s2*s4*s3*s4*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 37, 73)( 38, 74)( 39, 75)
( 40, 76)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 77)( 46, 78)( 47, 79)
( 48, 80)( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 93)( 54, 94)( 55, 95)
( 56, 96)( 57, 89)( 58, 90)( 59, 91)( 60, 92)( 61, 97)( 62, 98)( 63, 99)
( 64,100)( 65,105)( 66,106)( 67,107)( 68,108)( 69,101)( 70,102)( 71,103)
( 72,104)(113,117)(114,118)(115,119)(116,120)(125,129)(126,130)(127,131)
(128,132)(137,141)(138,142)(139,143)(140,144)(145,181)(146,182)(147,183)
(148,184)(149,189)(150,190)(151,191)(152,192)(153,185)(154,186)(155,187)
(156,188)(157,193)(158,194)(159,195)(160,196)(161,201)(162,202)(163,203)
(164,204)(165,197)(166,198)(167,199)(168,200)(169,205)(170,206)(171,207)
(172,208)(173,213)(174,214)(175,215)(176,216)(177,209)(178,210)(179,211)
(180,212)(221,225)(222,226)(223,227)(224,228)(233,237)(234,238)(235,239)
(236,240)(245,249)(246,250)(247,251)(248,252)(253,289)(254,290)(255,291)
(256,292)(257,297)(258,298)(259,299)(260,300)(261,293)(262,294)(263,295)
(264,296)(265,301)(266,302)(267,303)(268,304)(269,309)(270,310)(271,311)
(272,312)(273,305)(274,306)(275,307)(276,308)(277,313)(278,314)(279,315)
(280,316)(281,321)(282,322)(283,323)(284,324)(285,317)(286,318)(287,319)
(288,320);
s1 := 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);
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)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)
( 24,138)( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)
( 32,130)( 33,125)( 34,128)( 35,127)( 36,126)( 37,149)( 38,152)( 39,151)
( 40,150)( 41,145)( 42,148)( 43,147)( 44,146)( 45,153)( 46,156)( 47,155)
( 48,154)( 49,173)( 50,176)( 51,175)( 52,174)( 53,169)( 54,172)( 55,171)
( 56,170)( 57,177)( 58,180)( 59,179)( 60,178)( 61,161)( 62,164)( 63,163)
( 64,162)( 65,157)( 66,160)( 67,159)( 68,158)( 69,165)( 70,168)( 71,167)
( 72,166)( 73,189)( 74,192)( 75,191)( 76,190)( 77,185)( 78,188)( 79,187)
( 80,186)( 81,181)( 82,184)( 83,183)( 84,182)( 85,213)( 86,216)( 87,215)
( 88,214)( 89,209)( 90,212)( 91,211)( 92,210)( 93,205)( 94,208)( 95,207)
( 96,206)( 97,201)( 98,204)( 99,203)(100,202)(101,197)(102,200)(103,199)
(104,198)(105,193)(106,196)(107,195)(108,194)(217,241)(218,244)(219,243)
(220,242)(221,249)(222,252)(223,251)(224,250)(225,245)(226,248)(227,247)
(228,246)(230,232)(233,237)(234,240)(235,239)(236,238)(253,281)(254,284)
(255,283)(256,282)(257,277)(258,280)(259,279)(260,278)(261,285)(262,288)
(263,287)(264,286)(265,269)(266,272)(267,271)(268,270)(274,276)(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)!( 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);
s4 := 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);
poly := sub<Sym(324)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4,
s4*s3*s2*s4*s3*s4*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope