Polytope of Type {4,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,9}*1296a
if this polytope has a name.
Group : SmallGroup(1296,891)
Rank : 4
Schlafli Type : {4,6,9}
Number of vertices, edges, etc : 4, 36, 81, 27
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
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) :
2-fold quotients : {2,6,9}*648a
3-fold quotients : {4,6,9}*432, {4,6,3}*432a
6-fold quotients : {2,6,9}*216, {2,6,3}*216
9-fold quotients : {4,2,9}*144, {4,6,3}*144
18-fold quotients : {2,2,9}*72, {2,6,3}*72
27-fold quotients : {4,2,3}*48
54-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,163)( 2,164)( 3,165)( 4,166)( 5,167)( 6,168)( 7,169)( 8,170)
( 9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)
( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)
( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)
( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)
( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)
( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)
( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)
( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)
( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)
( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)
( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)
( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)
(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)
(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)
(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)
(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)
(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)
(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)
(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)
(161,323)(162,324);;
s1 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)(170,252)
(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)(178,268)
(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)(186,257)
(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)(194,276)
(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)(202,292)
(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)(210,281)
(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)(218,300)
(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)(226,316)
(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)(234,323)
(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)(242,315)
(243,314);;
s2 := ( 1, 10)( 2, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)( 8, 14)
( 9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 70)( 29, 71)( 30, 72)( 31, 67)
( 32, 68)( 33, 69)( 34, 64)( 35, 65)( 36, 66)( 37, 61)( 38, 62)( 39, 63)
( 40, 58)( 41, 59)( 42, 60)( 43, 55)( 44, 56)( 45, 57)( 46, 79)( 47, 80)
( 48, 81)( 49, 76)( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 82, 91)
( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)
(103,106)(104,107)(105,108)(109,151)(110,152)(111,153)(112,148)(113,149)
(114,150)(115,145)(116,146)(117,147)(118,142)(119,143)(120,144)(121,139)
(122,140)(123,141)(124,136)(125,137)(126,138)(127,160)(128,161)(129,162)
(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)(163,172)(164,173)
(165,174)(166,178)(167,179)(168,180)(169,175)(170,176)(171,177)(184,187)
(185,188)(186,189)(190,232)(191,233)(192,234)(193,229)(194,230)(195,231)
(196,226)(197,227)(198,228)(199,223)(200,224)(201,225)(202,220)(203,221)
(204,222)(205,217)(206,218)(207,219)(208,241)(209,242)(210,243)(211,238)
(212,239)(213,240)(214,235)(215,236)(216,237)(244,253)(245,254)(246,255)
(247,259)(248,260)(249,261)(250,256)(251,257)(252,258)(265,268)(266,269)
(267,270)(271,313)(272,314)(273,315)(274,310)(275,311)(276,312)(277,307)
(278,308)(279,309)(280,304)(281,305)(282,306)(283,301)(284,302)(285,303)
(286,298)(287,299)(288,300)(289,322)(290,323)(291,324)(292,319)(293,320)
(294,321)(295,316)(296,317)(297,318);;
s3 := ( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)( 8, 32)
( 9, 33)( 10, 48)( 11, 46)( 12, 47)( 13, 54)( 14, 52)( 15, 53)( 16, 51)
( 17, 49)( 18, 50)( 19, 38)( 20, 39)( 21, 37)( 22, 44)( 23, 45)( 24, 43)
( 25, 41)( 26, 42)( 27, 40)( 55, 61)( 56, 62)( 57, 63)( 64, 81)( 65, 79)
( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)( 82,109)
( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)
( 91,129)( 92,127)( 93,128)( 94,135)( 95,133)( 96,134)( 97,132)( 98,130)
( 99,131)(100,119)(101,120)(102,118)(103,125)(104,126)(105,124)(106,122)
(107,123)(108,121)(136,142)(137,143)(138,144)(145,162)(146,160)(147,161)
(148,159)(149,157)(150,158)(151,156)(152,154)(153,155)(163,190)(164,191)
(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,210)
(173,208)(174,209)(175,216)(176,214)(177,215)(178,213)(179,211)(180,212)
(181,200)(182,201)(183,199)(184,206)(185,207)(186,205)(187,203)(188,204)
(189,202)(217,223)(218,224)(219,225)(226,243)(227,241)(228,242)(229,240)
(230,238)(231,239)(232,237)(233,235)(234,236)(244,271)(245,272)(246,273)
(247,277)(248,278)(249,279)(250,274)(251,275)(252,276)(253,291)(254,289)
(255,290)(256,297)(257,295)(258,296)(259,294)(260,292)(261,293)(262,281)
(263,282)(264,280)(265,287)(266,288)(267,286)(268,284)(269,285)(270,283)
(298,304)(299,305)(300,306)(307,324)(308,322)(309,323)(310,321)(311,319)
(312,320)(313,318)(314,316)(315,317);;
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*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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)!( 1,163)( 2,164)( 3,165)( 4,166)( 5,167)( 6,168)( 7,169)
( 8,170)( 9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)
( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)
( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)
( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)
( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)
( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)
( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)
( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)
( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)
( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)
( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)
( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)
(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)
(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)
(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)
(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)
(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)
(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)
(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)
(160,322)(161,323)(162,324);
s1 := Sym(324)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)
(170,252)(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)
(178,268)(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)
(186,257)(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)
(194,276)(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)
(202,292)(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)
(210,281)(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)
(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)
(226,316)(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)
(234,323)(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)
(242,315)(243,314);
s2 := Sym(324)!( 1, 10)( 2, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)
( 8, 14)( 9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 70)( 29, 71)( 30, 72)
( 31, 67)( 32, 68)( 33, 69)( 34, 64)( 35, 65)( 36, 66)( 37, 61)( 38, 62)
( 39, 63)( 40, 58)( 41, 59)( 42, 60)( 43, 55)( 44, 56)( 45, 57)( 46, 79)
( 47, 80)( 48, 81)( 49, 76)( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)
( 82, 91)( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)
( 90, 96)(103,106)(104,107)(105,108)(109,151)(110,152)(111,153)(112,148)
(113,149)(114,150)(115,145)(116,146)(117,147)(118,142)(119,143)(120,144)
(121,139)(122,140)(123,141)(124,136)(125,137)(126,138)(127,160)(128,161)
(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)(163,172)
(164,173)(165,174)(166,178)(167,179)(168,180)(169,175)(170,176)(171,177)
(184,187)(185,188)(186,189)(190,232)(191,233)(192,234)(193,229)(194,230)
(195,231)(196,226)(197,227)(198,228)(199,223)(200,224)(201,225)(202,220)
(203,221)(204,222)(205,217)(206,218)(207,219)(208,241)(209,242)(210,243)
(211,238)(212,239)(213,240)(214,235)(215,236)(216,237)(244,253)(245,254)
(246,255)(247,259)(248,260)(249,261)(250,256)(251,257)(252,258)(265,268)
(266,269)(267,270)(271,313)(272,314)(273,315)(274,310)(275,311)(276,312)
(277,307)(278,308)(279,309)(280,304)(281,305)(282,306)(283,301)(284,302)
(285,303)(286,298)(287,299)(288,300)(289,322)(290,323)(291,324)(292,319)
(293,320)(294,321)(295,316)(296,317)(297,318);
s3 := Sym(324)!( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)
( 8, 32)( 9, 33)( 10, 48)( 11, 46)( 12, 47)( 13, 54)( 14, 52)( 15, 53)
( 16, 51)( 17, 49)( 18, 50)( 19, 38)( 20, 39)( 21, 37)( 22, 44)( 23, 45)
( 24, 43)( 25, 41)( 26, 42)( 27, 40)( 55, 61)( 56, 62)( 57, 63)( 64, 81)
( 65, 79)( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)
( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)
( 90,114)( 91,129)( 92,127)( 93,128)( 94,135)( 95,133)( 96,134)( 97,132)
( 98,130)( 99,131)(100,119)(101,120)(102,118)(103,125)(104,126)(105,124)
(106,122)(107,123)(108,121)(136,142)(137,143)(138,144)(145,162)(146,160)
(147,161)(148,159)(149,157)(150,158)(151,156)(152,154)(153,155)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,210)(173,208)(174,209)(175,216)(176,214)(177,215)(178,213)(179,211)
(180,212)(181,200)(182,201)(183,199)(184,206)(185,207)(186,205)(187,203)
(188,204)(189,202)(217,223)(218,224)(219,225)(226,243)(227,241)(228,242)
(229,240)(230,238)(231,239)(232,237)(233,235)(234,236)(244,271)(245,272)
(246,273)(247,277)(248,278)(249,279)(250,274)(251,275)(252,276)(253,291)
(254,289)(255,290)(256,297)(257,295)(258,296)(259,294)(260,292)(261,293)
(262,281)(263,282)(264,280)(265,287)(266,288)(267,286)(268,284)(269,285)
(270,283)(298,304)(299,305)(300,306)(307,324)(308,322)(309,323)(310,321)
(311,319)(312,320)(313,318)(314,316)(315,317);
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*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope