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