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