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