Polytope of Type {4,18,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,9}*1296
if this polytope has a name.
Group : SmallGroup(1296,878)
Rank : 4
Schlafli Type : {4,18,9}
Number of vertices, edges, etc : 4, 36, 81, 9
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 : {2,18,9}*648
3-fold quotients : {4,6,9}*432
6-fold quotients : {2,6,9}*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 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)
( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)( 42, 69)
( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)
( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)(111,144)
(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)(119,152)
(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(127,160)
(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)
(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)
(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)
(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)
(187,265)(188,266)(189,267)(190,304)(191,305)(192,306)(193,301)(194,302)
(195,303)(196,298)(197,299)(198,300)(199,313)(200,314)(201,315)(202,310)
(203,311)(204,312)(205,307)(206,308)(207,309)(208,322)(209,323)(210,324)
(211,319)(212,320)(213,321)(214,316)(215,317)(216,318)(217,277)(218,278)
(219,279)(220,274)(221,275)(222,276)(223,271)(224,272)(225,273)(226,286)
(227,287)(228,288)(229,283)(230,284)(231,285)(232,280)(233,281)(234,282)
(235,295)(236,296)(237,297)(238,292)(239,293)(240,294)(241,289)(242,290)
(243,291);;
s2 := ( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)( 8, 33)
( 9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 55, 61)( 56, 63)( 57, 62)( 59, 60)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,129)( 92,128)( 93,127)( 94,135)( 95,134)( 96,133)( 97,132)
( 98,131)( 99,130)(100,120)(101,119)(102,118)(103,126)(104,125)(105,124)
(106,123)(107,122)(108,121)(136,142)(137,144)(138,143)(140,141)(145,162)
(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)
(163,190)(164,192)(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)
(171,194)(172,210)(173,209)(174,208)(175,216)(176,215)(177,214)(178,213)
(179,212)(180,211)(181,201)(182,200)(183,199)(184,207)(185,206)(186,205)
(187,204)(188,203)(189,202)(217,223)(218,225)(219,224)(221,222)(226,243)
(227,242)(228,241)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235)
(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)
(252,275)(253,291)(254,290)(255,289)(256,297)(257,296)(258,295)(259,294)
(260,293)(261,292)(262,282)(263,281)(264,280)(265,288)(266,287)(267,286)
(268,285)(269,284)(270,283)(298,304)(299,306)(300,305)(302,303)(307,324)
(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)(314,317)(315,316);;
s3 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 70)( 29, 72)( 30, 71)
( 31, 67)( 32, 69)( 33, 68)( 34, 64)( 35, 66)( 36, 65)( 37, 61)( 38, 63)
( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 81)
( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,151)(110,153)(111,152)
(112,148)(113,150)(114,149)(115,145)(116,147)(117,146)(118,142)(119,144)
(120,143)(121,139)(122,141)(123,140)(124,136)(125,138)(126,137)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(163,172)(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)
(171,176)(181,183)(184,189)(185,188)(186,187)(190,232)(191,234)(192,233)
(193,229)(194,231)(195,230)(196,226)(197,228)(198,227)(199,223)(200,225)
(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,243)
(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)
(244,253)(245,255)(246,254)(247,259)(248,261)(249,260)(250,256)(251,258)
(252,257)(262,264)(265,270)(266,269)(267,268)(271,313)(272,315)(273,314)
(274,310)(275,312)(276,311)(277,307)(278,309)(279,308)(280,304)(281,306)
(282,305)(283,301)(284,303)(285,302)(286,298)(287,300)(288,299)(289,324)
(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)(297,316);;
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, 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)!( 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)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)
( 34, 55)( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)
( 42, 69)( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)
( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)
(111,144)(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)
(119,152)(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)
(127,160)(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)
(135,156)(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)
(170,248)(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)
(178,256)(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)
(186,270)(187,265)(188,266)(189,267)(190,304)(191,305)(192,306)(193,301)
(194,302)(195,303)(196,298)(197,299)(198,300)(199,313)(200,314)(201,315)
(202,310)(203,311)(204,312)(205,307)(206,308)(207,309)(208,322)(209,323)
(210,324)(211,319)(212,320)(213,321)(214,316)(215,317)(216,318)(217,277)
(218,278)(219,279)(220,274)(221,275)(222,276)(223,271)(224,272)(225,273)
(226,286)(227,287)(228,288)(229,283)(230,284)(231,285)(232,280)(233,281)
(234,282)(235,295)(236,296)(237,297)(238,292)(239,293)(240,294)(241,289)
(242,290)(243,291);
s2 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)
( 8, 33)( 9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 61)( 56, 63)( 57, 62)( 59, 60)
( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)
( 72, 73)( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)
( 89,114)( 90,113)( 91,129)( 92,128)( 93,127)( 94,135)( 95,134)( 96,133)
( 97,132)( 98,131)( 99,130)(100,120)(101,119)(102,118)(103,126)(104,125)
(105,124)(106,123)(107,122)(108,121)(136,142)(137,144)(138,143)(140,141)
(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)
(153,154)(163,190)(164,192)(165,191)(166,196)(167,198)(168,197)(169,193)
(170,195)(171,194)(172,210)(173,209)(174,208)(175,216)(176,215)(177,214)
(178,213)(179,212)(180,211)(181,201)(182,200)(183,199)(184,207)(185,206)
(186,205)(187,204)(188,203)(189,202)(217,223)(218,225)(219,224)(221,222)
(226,243)(227,242)(228,241)(229,240)(230,239)(231,238)(232,237)(233,236)
(234,235)(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)
(251,276)(252,275)(253,291)(254,290)(255,289)(256,297)(257,296)(258,295)
(259,294)(260,293)(261,292)(262,282)(263,281)(264,280)(265,288)(266,287)
(267,286)(268,285)(269,284)(270,283)(298,304)(299,306)(300,305)(302,303)
(307,324)(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)(314,317)
(315,316);
s3 := Sym(324)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 70)( 29, 72)
( 30, 71)( 31, 67)( 32, 69)( 33, 68)( 34, 64)( 35, 66)( 36, 65)( 37, 61)
( 38, 63)( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)
( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)
( 54, 73)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)
( 89, 96)( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,151)(110,153)
(111,152)(112,148)(113,150)(114,149)(115,145)(116,147)(117,146)(118,142)
(119,144)(120,143)(121,139)(122,141)(123,140)(124,136)(125,138)(126,137)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)
(135,154)(163,172)(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)
(170,177)(171,176)(181,183)(184,189)(185,188)(186,187)(190,232)(191,234)
(192,233)(193,229)(194,231)(195,230)(196,226)(197,228)(198,227)(199,223)
(200,225)(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)
(208,243)(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)
(216,235)(244,253)(245,255)(246,254)(247,259)(248,261)(249,260)(250,256)
(251,258)(252,257)(262,264)(265,270)(266,269)(267,268)(271,313)(272,315)
(273,314)(274,310)(275,312)(276,311)(277,307)(278,309)(279,308)(280,304)
(281,306)(282,305)(283,301)(284,303)(285,302)(286,298)(287,300)(288,299)
(289,324)(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)
(297,316);
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,
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