Polytope of Type {4,18,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,10}*1440b
if this polytope has a name.
Group : SmallGroup(1440,4569)
Rank : 4
Schlafli Type : {4,18,10}
Number of vertices, edges, etc : 4, 36, 90, 10
Order of s0s1s2s3 : 90
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 : {4,6,10}*480b
5-fold quotients : {4,18,2}*288b
10-fold quotients : {4,9,2}*144
15-fold quotients : {4,6,2}*96c
30-fold quotients : {4,3,2}*48
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)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)(350,352)
(353,355)(354,356)(357,359)(358,360);;
s1 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 61,129)( 62,130)( 63,132)( 64,131)( 65,125)( 66,126)( 67,128)
( 68,127)( 69,121)( 70,122)( 71,124)( 72,123)( 73,141)( 74,142)( 75,144)
( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,133)( 82,134)( 83,136)
( 84,135)( 85,153)( 86,154)( 87,156)( 88,155)( 89,149)( 90,150)( 91,152)
( 92,151)( 93,145)( 94,146)( 95,148)( 96,147)( 97,165)( 98,166)( 99,168)
(100,167)(101,161)(102,162)(103,164)(104,163)(105,157)(106,158)(107,160)
(108,159)(109,177)(110,178)(111,180)(112,179)(113,173)(114,174)(115,176)
(116,175)(117,169)(118,170)(119,172)(120,171)(183,184)(185,189)(186,190)
(187,192)(188,191)(195,196)(197,201)(198,202)(199,204)(200,203)(207,208)
(209,213)(210,214)(211,216)(212,215)(219,220)(221,225)(222,226)(223,228)
(224,227)(231,232)(233,237)(234,238)(235,240)(236,239)(241,309)(242,310)
(243,312)(244,311)(245,305)(246,306)(247,308)(248,307)(249,301)(250,302)
(251,304)(252,303)(253,321)(254,322)(255,324)(256,323)(257,317)(258,318)
(259,320)(260,319)(261,313)(262,314)(263,316)(264,315)(265,333)(266,334)
(267,336)(268,335)(269,329)(270,330)(271,332)(272,331)(273,325)(274,326)
(275,328)(276,327)(277,345)(278,346)(279,348)(280,347)(281,341)(282,342)
(283,344)(284,343)(285,337)(286,338)(287,340)(288,339)(289,357)(290,358)
(291,360)(292,359)(293,353)(294,354)(295,356)(296,355)(297,349)(298,350)
(299,352)(300,351);;
s2 := ( 1, 61)( 2, 64)( 3, 63)( 4, 62)( 5, 69)( 6, 72)( 7, 71)( 8, 70)
( 9, 65)( 10, 68)( 11, 67)( 12, 66)( 13,109)( 14,112)( 15,111)( 16,110)
( 17,117)( 18,120)( 19,119)( 20,118)( 21,113)( 22,116)( 23,115)( 24,114)
( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)( 32,106)
( 33,101)( 34,104)( 35,103)( 36,102)( 37, 85)( 38, 88)( 39, 87)( 40, 86)
( 41, 93)( 42, 96)( 43, 95)( 44, 94)( 45, 89)( 46, 92)( 47, 91)( 48, 90)
( 49, 73)( 50, 76)( 51, 75)( 52, 74)( 53, 81)( 54, 84)( 55, 83)( 56, 82)
( 57, 77)( 58, 80)( 59, 79)( 60, 78)(121,129)(122,132)(123,131)(124,130)
(126,128)(133,177)(134,180)(135,179)(136,178)(137,173)(138,176)(139,175)
(140,174)(141,169)(142,172)(143,171)(144,170)(145,165)(146,168)(147,167)
(148,166)(149,161)(150,164)(151,163)(152,162)(153,157)(154,160)(155,159)
(156,158)(181,241)(182,244)(183,243)(184,242)(185,249)(186,252)(187,251)
(188,250)(189,245)(190,248)(191,247)(192,246)(193,289)(194,292)(195,291)
(196,290)(197,297)(198,300)(199,299)(200,298)(201,293)(202,296)(203,295)
(204,294)(205,277)(206,280)(207,279)(208,278)(209,285)(210,288)(211,287)
(212,286)(213,281)(214,284)(215,283)(216,282)(217,265)(218,268)(219,267)
(220,266)(221,273)(222,276)(223,275)(224,274)(225,269)(226,272)(227,271)
(228,270)(229,253)(230,256)(231,255)(232,254)(233,261)(234,264)(235,263)
(236,262)(237,257)(238,260)(239,259)(240,258)(301,309)(302,312)(303,311)
(304,310)(306,308)(313,357)(314,360)(315,359)(316,358)(317,353)(318,356)
(319,355)(320,354)(321,349)(322,352)(323,351)(324,350)(325,345)(326,348)
(327,347)(328,346)(329,341)(330,344)(331,343)(332,342)(333,337)(334,340)
(335,339)(336,338);;
s3 := ( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,199)( 8,200)
( 9,201)( 10,202)( 11,203)( 12,204)( 13,181)( 14,182)( 15,183)( 16,184)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,229)( 26,230)( 27,231)( 28,232)( 29,233)( 30,234)( 31,235)( 32,236)
( 33,237)( 34,238)( 35,239)( 36,240)( 37,217)( 38,218)( 39,219)( 40,220)
( 41,221)( 42,222)( 43,223)( 44,224)( 45,225)( 46,226)( 47,227)( 48,228)
( 49,205)( 50,206)( 51,207)( 52,208)( 53,209)( 54,210)( 55,211)( 56,212)
( 57,213)( 58,214)( 59,215)( 60,216)( 61,253)( 62,254)( 63,255)( 64,256)
( 65,257)( 66,258)( 67,259)( 68,260)( 69,261)( 70,262)( 71,263)( 72,264)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)( 80,248)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,289)( 86,290)( 87,291)( 88,292)
( 89,293)( 90,294)( 91,295)( 92,296)( 93,297)( 94,298)( 95,299)( 96,300)
( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,283)(104,284)
(105,285)(106,286)(107,287)(108,288)(109,265)(110,266)(111,267)(112,268)
(113,269)(114,270)(115,271)(116,272)(117,273)(118,274)(119,275)(120,276)
(121,313)(122,314)(123,315)(124,316)(125,317)(126,318)(127,319)(128,320)
(129,321)(130,322)(131,323)(132,324)(133,301)(134,302)(135,303)(136,304)
(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)(144,312)
(145,349)(146,350)(147,351)(148,352)(149,353)(150,354)(151,355)(152,356)
(153,357)(154,358)(155,359)(156,360)(157,337)(158,338)(159,339)(160,340)
(161,341)(162,342)(163,343)(164,344)(165,345)(166,346)(167,347)(168,348)
(169,325)(170,326)(171,327)(172,328)(173,329)(174,330)(175,331)(176,332)
(177,333)(178,334)(179,335)(180,336);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(360)!( 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)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)
(350,352)(353,355)(354,356)(357,359)(358,360);
s1 := Sym(360)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 61,129)( 62,130)( 63,132)( 64,131)( 65,125)( 66,126)
( 67,128)( 68,127)( 69,121)( 70,122)( 71,124)( 72,123)( 73,141)( 74,142)
( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,133)( 82,134)
( 83,136)( 84,135)( 85,153)( 86,154)( 87,156)( 88,155)( 89,149)( 90,150)
( 91,152)( 92,151)( 93,145)( 94,146)( 95,148)( 96,147)( 97,165)( 98,166)
( 99,168)(100,167)(101,161)(102,162)(103,164)(104,163)(105,157)(106,158)
(107,160)(108,159)(109,177)(110,178)(111,180)(112,179)(113,173)(114,174)
(115,176)(116,175)(117,169)(118,170)(119,172)(120,171)(183,184)(185,189)
(186,190)(187,192)(188,191)(195,196)(197,201)(198,202)(199,204)(200,203)
(207,208)(209,213)(210,214)(211,216)(212,215)(219,220)(221,225)(222,226)
(223,228)(224,227)(231,232)(233,237)(234,238)(235,240)(236,239)(241,309)
(242,310)(243,312)(244,311)(245,305)(246,306)(247,308)(248,307)(249,301)
(250,302)(251,304)(252,303)(253,321)(254,322)(255,324)(256,323)(257,317)
(258,318)(259,320)(260,319)(261,313)(262,314)(263,316)(264,315)(265,333)
(266,334)(267,336)(268,335)(269,329)(270,330)(271,332)(272,331)(273,325)
(274,326)(275,328)(276,327)(277,345)(278,346)(279,348)(280,347)(281,341)
(282,342)(283,344)(284,343)(285,337)(286,338)(287,340)(288,339)(289,357)
(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)(296,355)(297,349)
(298,350)(299,352)(300,351);
s2 := Sym(360)!( 1, 61)( 2, 64)( 3, 63)( 4, 62)( 5, 69)( 6, 72)( 7, 71)
( 8, 70)( 9, 65)( 10, 68)( 11, 67)( 12, 66)( 13,109)( 14,112)( 15,111)
( 16,110)( 17,117)( 18,120)( 19,119)( 20,118)( 21,113)( 22,116)( 23,115)
( 24,114)( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)
( 32,106)( 33,101)( 34,104)( 35,103)( 36,102)( 37, 85)( 38, 88)( 39, 87)
( 40, 86)( 41, 93)( 42, 96)( 43, 95)( 44, 94)( 45, 89)( 46, 92)( 47, 91)
( 48, 90)( 49, 73)( 50, 76)( 51, 75)( 52, 74)( 53, 81)( 54, 84)( 55, 83)
( 56, 82)( 57, 77)( 58, 80)( 59, 79)( 60, 78)(121,129)(122,132)(123,131)
(124,130)(126,128)(133,177)(134,180)(135,179)(136,178)(137,173)(138,176)
(139,175)(140,174)(141,169)(142,172)(143,171)(144,170)(145,165)(146,168)
(147,167)(148,166)(149,161)(150,164)(151,163)(152,162)(153,157)(154,160)
(155,159)(156,158)(181,241)(182,244)(183,243)(184,242)(185,249)(186,252)
(187,251)(188,250)(189,245)(190,248)(191,247)(192,246)(193,289)(194,292)
(195,291)(196,290)(197,297)(198,300)(199,299)(200,298)(201,293)(202,296)
(203,295)(204,294)(205,277)(206,280)(207,279)(208,278)(209,285)(210,288)
(211,287)(212,286)(213,281)(214,284)(215,283)(216,282)(217,265)(218,268)
(219,267)(220,266)(221,273)(222,276)(223,275)(224,274)(225,269)(226,272)
(227,271)(228,270)(229,253)(230,256)(231,255)(232,254)(233,261)(234,264)
(235,263)(236,262)(237,257)(238,260)(239,259)(240,258)(301,309)(302,312)
(303,311)(304,310)(306,308)(313,357)(314,360)(315,359)(316,358)(317,353)
(318,356)(319,355)(320,354)(321,349)(322,352)(323,351)(324,350)(325,345)
(326,348)(327,347)(328,346)(329,341)(330,344)(331,343)(332,342)(333,337)
(334,340)(335,339)(336,338);
s3 := Sym(360)!( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,199)
( 8,200)( 9,201)( 10,202)( 11,203)( 12,204)( 13,181)( 14,182)( 15,183)
( 16,184)( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)
( 24,192)( 25,229)( 26,230)( 27,231)( 28,232)( 29,233)( 30,234)( 31,235)
( 32,236)( 33,237)( 34,238)( 35,239)( 36,240)( 37,217)( 38,218)( 39,219)
( 40,220)( 41,221)( 42,222)( 43,223)( 44,224)( 45,225)( 46,226)( 47,227)
( 48,228)( 49,205)( 50,206)( 51,207)( 52,208)( 53,209)( 54,210)( 55,211)
( 56,212)( 57,213)( 58,214)( 59,215)( 60,216)( 61,253)( 62,254)( 63,255)
( 64,256)( 65,257)( 66,258)( 67,259)( 68,260)( 69,261)( 70,262)( 71,263)
( 72,264)( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)
( 80,248)( 81,249)( 82,250)( 83,251)( 84,252)( 85,289)( 86,290)( 87,291)
( 88,292)( 89,293)( 90,294)( 91,295)( 92,296)( 93,297)( 94,298)( 95,299)
( 96,300)( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,283)
(104,284)(105,285)(106,286)(107,287)(108,288)(109,265)(110,266)(111,267)
(112,268)(113,269)(114,270)(115,271)(116,272)(117,273)(118,274)(119,275)
(120,276)(121,313)(122,314)(123,315)(124,316)(125,317)(126,318)(127,319)
(128,320)(129,321)(130,322)(131,323)(132,324)(133,301)(134,302)(135,303)
(136,304)(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)
(144,312)(145,349)(146,350)(147,351)(148,352)(149,353)(150,354)(151,355)
(152,356)(153,357)(154,358)(155,359)(156,360)(157,337)(158,338)(159,339)
(160,340)(161,341)(162,342)(163,343)(164,344)(165,345)(166,346)(167,347)
(168,348)(169,325)(170,326)(171,327)(172,328)(173,329)(174,330)(175,331)
(176,332)(177,333)(178,334)(179,335)(180,336);
poly := sub<Sym(360)|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, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope