Polytope of Type {5,10,15}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10,15}*1500
if this polytope has a name.
Group : SmallGroup(1500,72)
Rank : 4
Schlafli Type : {5,10,15}
Number of vertices, edges, etc : 5, 25, 75, 15
Order of s0s1s2s3 : 15
Order of s0s1s2s3s2s1 : 10
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 : {5,10,5}*500
5-fold quotients : {5,2,15}*300
15-fold quotients : {5,2,5}*100
25-fold quotients : {5,2,3}*60
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 4)( 6, 21)( 7, 25)( 8, 24)( 9, 23)( 10, 22)( 11, 16)
( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 27, 30)( 28, 29)( 31, 46)( 32, 50)
( 33, 49)( 34, 48)( 35, 47)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)
( 52, 55)( 53, 54)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 66)
( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 77, 80)( 78, 79)( 81, 96)( 82,100)
( 83, 99)( 84, 98)( 85, 97)( 86, 91)( 87, 95)( 88, 94)( 89, 93)( 90, 92)
(102,105)(103,104)(106,121)(107,125)(108,124)(109,123)(110,122)(111,116)
(112,120)(113,119)(114,118)(115,117)(127,130)(128,129)(131,146)(132,150)
(133,149)(134,148)(135,147)(136,141)(137,145)(138,144)(139,143)(140,142)
(152,155)(153,154)(156,171)(157,175)(158,174)(159,173)(160,172)(161,166)
(162,170)(163,169)(164,168)(165,167)(177,180)(178,179)(181,196)(182,200)
(183,199)(184,198)(185,197)(186,191)(187,195)(188,194)(189,193)(190,192)
(202,205)(203,204)(206,221)(207,225)(208,224)(209,223)(210,222)(211,216)
(212,220)(213,219)(214,218)(215,217)(227,230)(228,229)(231,246)(232,250)
(233,249)(234,248)(235,247)(236,241)(237,245)(238,244)(239,243)(240,242)
(252,255)(253,254)(256,271)(257,275)(258,274)(259,273)(260,272)(261,266)
(262,270)(263,269)(264,268)(265,267)(277,280)(278,279)(281,296)(282,300)
(283,299)(284,298)(285,297)(286,291)(287,295)(288,294)(289,293)(290,292)
(302,305)(303,304)(306,321)(307,325)(308,324)(309,323)(310,322)(311,316)
(312,320)(313,319)(314,318)(315,317)(327,330)(328,329)(331,346)(332,350)
(333,349)(334,348)(335,347)(336,341)(337,345)(338,344)(339,343)(340,342)
(352,355)(353,354)(356,371)(357,375)(358,374)(359,373)(360,372)(361,366)
(362,370)(363,369)(364,368)(365,367);;
s1 := ( 1, 6)( 2, 10)( 3, 9)( 4, 8)( 5, 7)( 11, 21)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)
( 30, 32)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 42, 45)( 43, 44)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 71)( 62, 75)( 63, 74)
( 64, 73)( 65, 72)( 67, 70)( 68, 69)( 76, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)( 93, 94)
(101,106)(102,110)(103,109)(104,108)(105,107)(111,121)(112,125)(113,124)
(114,123)(115,122)(117,120)(118,119)(126,131)(127,135)(128,134)(129,133)
(130,132)(136,146)(137,150)(138,149)(139,148)(140,147)(142,145)(143,144)
(151,156)(152,160)(153,159)(154,158)(155,157)(161,171)(162,175)(163,174)
(164,173)(165,172)(167,170)(168,169)(176,181)(177,185)(178,184)(179,183)
(180,182)(186,196)(187,200)(188,199)(189,198)(190,197)(192,195)(193,194)
(201,206)(202,210)(203,209)(204,208)(205,207)(211,221)(212,225)(213,224)
(214,223)(215,222)(217,220)(218,219)(226,231)(227,235)(228,234)(229,233)
(230,232)(236,246)(237,250)(238,249)(239,248)(240,247)(242,245)(243,244)
(251,256)(252,260)(253,259)(254,258)(255,257)(261,271)(262,275)(263,274)
(264,273)(265,272)(267,270)(268,269)(276,281)(277,285)(278,284)(279,283)
(280,282)(286,296)(287,300)(288,299)(289,298)(290,297)(292,295)(293,294)
(301,306)(302,310)(303,309)(304,308)(305,307)(311,321)(312,325)(313,324)
(314,323)(315,322)(317,320)(318,319)(326,331)(327,335)(328,334)(329,333)
(330,332)(336,346)(337,350)(338,349)(339,348)(340,347)(342,345)(343,344)
(351,356)(352,360)(353,359)(354,358)(355,357)(361,371)(362,375)(363,374)
(364,373)(365,372)(367,370)(368,369);;
s2 := ( 1, 26)( 2, 30)( 3, 29)( 4, 28)( 5, 27)( 6, 32)( 7, 31)( 8, 35)
( 9, 34)( 10, 33)( 11, 38)( 12, 37)( 13, 36)( 14, 40)( 15, 39)( 16, 44)
( 17, 43)( 18, 42)( 19, 41)( 20, 45)( 21, 50)( 22, 49)( 23, 48)( 24, 47)
( 25, 46)( 51,101)( 52,105)( 53,104)( 54,103)( 55,102)( 56,107)( 57,106)
( 58,110)( 59,109)( 60,108)( 61,113)( 62,112)( 63,111)( 64,115)( 65,114)
( 66,119)( 67,118)( 68,117)( 69,116)( 70,120)( 71,125)( 72,124)( 73,123)
( 74,122)( 75,121)( 77, 80)( 78, 79)( 81, 82)( 83, 85)( 86, 88)( 89, 90)
( 91, 94)( 92, 93)( 96,100)( 97, 99)(126,276)(127,280)(128,279)(129,278)
(130,277)(131,282)(132,281)(133,285)(134,284)(135,283)(136,288)(137,287)
(138,286)(139,290)(140,289)(141,294)(142,293)(143,292)(144,291)(145,295)
(146,300)(147,299)(148,298)(149,297)(150,296)(151,251)(152,255)(153,254)
(154,253)(155,252)(156,257)(157,256)(158,260)(159,259)(160,258)(161,263)
(162,262)(163,261)(164,265)(165,264)(166,269)(167,268)(168,267)(169,266)
(170,270)(171,275)(172,274)(173,273)(174,272)(175,271)(176,351)(177,355)
(178,354)(179,353)(180,352)(181,357)(182,356)(183,360)(184,359)(185,358)
(186,363)(187,362)(188,361)(189,365)(190,364)(191,369)(192,368)(193,367)
(194,366)(195,370)(196,375)(197,374)(198,373)(199,372)(200,371)(201,326)
(202,330)(203,329)(204,328)(205,327)(206,332)(207,331)(208,335)(209,334)
(210,333)(211,338)(212,337)(213,336)(214,340)(215,339)(216,344)(217,343)
(218,342)(219,341)(220,345)(221,350)(222,349)(223,348)(224,347)(225,346)
(226,301)(227,305)(228,304)(229,303)(230,302)(231,307)(232,306)(233,310)
(234,309)(235,308)(236,313)(237,312)(238,311)(239,315)(240,314)(241,319)
(242,318)(243,317)(244,316)(245,320)(246,325)(247,324)(248,323)(249,322)
(250,321);;
s3 := ( 1,126)( 2,130)( 3,129)( 4,128)( 5,127)( 6,131)( 7,135)( 8,134)
( 9,133)( 10,132)( 11,136)( 12,140)( 13,139)( 14,138)( 15,137)( 16,141)
( 17,145)( 18,144)( 19,143)( 20,142)( 21,146)( 22,150)( 23,149)( 24,148)
( 25,147)( 26,226)( 27,230)( 28,229)( 29,228)( 30,227)( 31,231)( 32,235)
( 33,234)( 34,233)( 35,232)( 36,236)( 37,240)( 38,239)( 39,238)( 40,237)
( 41,241)( 42,245)( 43,244)( 44,243)( 45,242)( 46,246)( 47,250)( 48,249)
( 49,248)( 50,247)( 51,201)( 52,205)( 53,204)( 54,203)( 55,202)( 56,206)
( 57,210)( 58,209)( 59,208)( 60,207)( 61,211)( 62,215)( 63,214)( 64,213)
( 65,212)( 66,216)( 67,220)( 68,219)( 69,218)( 70,217)( 71,221)( 72,225)
( 73,224)( 74,223)( 75,222)( 76,176)( 77,180)( 78,179)( 79,178)( 80,177)
( 81,181)( 82,185)( 83,184)( 84,183)( 85,182)( 86,186)( 87,190)( 88,189)
( 89,188)( 90,187)( 91,191)( 92,195)( 93,194)( 94,193)( 95,192)( 96,196)
( 97,200)( 98,199)( 99,198)(100,197)(101,151)(102,155)(103,154)(104,153)
(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,161)(112,165)
(113,164)(114,163)(115,162)(116,166)(117,170)(118,169)(119,168)(120,167)
(121,171)(122,175)(123,174)(124,173)(125,172)(252,255)(253,254)(257,260)
(258,259)(262,265)(263,264)(267,270)(268,269)(272,275)(273,274)(276,351)
(277,355)(278,354)(279,353)(280,352)(281,356)(282,360)(283,359)(284,358)
(285,357)(286,361)(287,365)(288,364)(289,363)(290,362)(291,366)(292,370)
(293,369)(294,368)(295,367)(296,371)(297,375)(298,374)(299,373)(300,372)
(301,326)(302,330)(303,329)(304,328)(305,327)(306,331)(307,335)(308,334)
(309,333)(310,332)(311,336)(312,340)(313,339)(314,338)(315,337)(316,341)
(317,345)(318,344)(319,343)(320,342)(321,346)(322,350)(323,349)(324,348)
(325,347);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(375)!( 2, 5)( 3, 4)( 6, 21)( 7, 25)( 8, 24)( 9, 23)( 10, 22)
( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 27, 30)( 28, 29)( 31, 46)
( 32, 50)( 33, 49)( 34, 48)( 35, 47)( 36, 41)( 37, 45)( 38, 44)( 39, 43)
( 40, 42)( 52, 55)( 53, 54)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)
( 61, 66)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 77, 80)( 78, 79)( 81, 96)
( 82,100)( 83, 99)( 84, 98)( 85, 97)( 86, 91)( 87, 95)( 88, 94)( 89, 93)
( 90, 92)(102,105)(103,104)(106,121)(107,125)(108,124)(109,123)(110,122)
(111,116)(112,120)(113,119)(114,118)(115,117)(127,130)(128,129)(131,146)
(132,150)(133,149)(134,148)(135,147)(136,141)(137,145)(138,144)(139,143)
(140,142)(152,155)(153,154)(156,171)(157,175)(158,174)(159,173)(160,172)
(161,166)(162,170)(163,169)(164,168)(165,167)(177,180)(178,179)(181,196)
(182,200)(183,199)(184,198)(185,197)(186,191)(187,195)(188,194)(189,193)
(190,192)(202,205)(203,204)(206,221)(207,225)(208,224)(209,223)(210,222)
(211,216)(212,220)(213,219)(214,218)(215,217)(227,230)(228,229)(231,246)
(232,250)(233,249)(234,248)(235,247)(236,241)(237,245)(238,244)(239,243)
(240,242)(252,255)(253,254)(256,271)(257,275)(258,274)(259,273)(260,272)
(261,266)(262,270)(263,269)(264,268)(265,267)(277,280)(278,279)(281,296)
(282,300)(283,299)(284,298)(285,297)(286,291)(287,295)(288,294)(289,293)
(290,292)(302,305)(303,304)(306,321)(307,325)(308,324)(309,323)(310,322)
(311,316)(312,320)(313,319)(314,318)(315,317)(327,330)(328,329)(331,346)
(332,350)(333,349)(334,348)(335,347)(336,341)(337,345)(338,344)(339,343)
(340,342)(352,355)(353,354)(356,371)(357,375)(358,374)(359,373)(360,372)
(361,366)(362,370)(363,369)(364,368)(365,367);
s1 := Sym(375)!( 1, 6)( 2, 10)( 3, 9)( 4, 8)( 5, 7)( 11, 21)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)
( 29, 33)( 30, 32)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 42, 45)
( 43, 44)( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 71)( 62, 75)
( 63, 74)( 64, 73)( 65, 72)( 67, 70)( 68, 69)( 76, 81)( 77, 85)( 78, 84)
( 79, 83)( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)
( 93, 94)(101,106)(102,110)(103,109)(104,108)(105,107)(111,121)(112,125)
(113,124)(114,123)(115,122)(117,120)(118,119)(126,131)(127,135)(128,134)
(129,133)(130,132)(136,146)(137,150)(138,149)(139,148)(140,147)(142,145)
(143,144)(151,156)(152,160)(153,159)(154,158)(155,157)(161,171)(162,175)
(163,174)(164,173)(165,172)(167,170)(168,169)(176,181)(177,185)(178,184)
(179,183)(180,182)(186,196)(187,200)(188,199)(189,198)(190,197)(192,195)
(193,194)(201,206)(202,210)(203,209)(204,208)(205,207)(211,221)(212,225)
(213,224)(214,223)(215,222)(217,220)(218,219)(226,231)(227,235)(228,234)
(229,233)(230,232)(236,246)(237,250)(238,249)(239,248)(240,247)(242,245)
(243,244)(251,256)(252,260)(253,259)(254,258)(255,257)(261,271)(262,275)
(263,274)(264,273)(265,272)(267,270)(268,269)(276,281)(277,285)(278,284)
(279,283)(280,282)(286,296)(287,300)(288,299)(289,298)(290,297)(292,295)
(293,294)(301,306)(302,310)(303,309)(304,308)(305,307)(311,321)(312,325)
(313,324)(314,323)(315,322)(317,320)(318,319)(326,331)(327,335)(328,334)
(329,333)(330,332)(336,346)(337,350)(338,349)(339,348)(340,347)(342,345)
(343,344)(351,356)(352,360)(353,359)(354,358)(355,357)(361,371)(362,375)
(363,374)(364,373)(365,372)(367,370)(368,369);
s2 := Sym(375)!( 1, 26)( 2, 30)( 3, 29)( 4, 28)( 5, 27)( 6, 32)( 7, 31)
( 8, 35)( 9, 34)( 10, 33)( 11, 38)( 12, 37)( 13, 36)( 14, 40)( 15, 39)
( 16, 44)( 17, 43)( 18, 42)( 19, 41)( 20, 45)( 21, 50)( 22, 49)( 23, 48)
( 24, 47)( 25, 46)( 51,101)( 52,105)( 53,104)( 54,103)( 55,102)( 56,107)
( 57,106)( 58,110)( 59,109)( 60,108)( 61,113)( 62,112)( 63,111)( 64,115)
( 65,114)( 66,119)( 67,118)( 68,117)( 69,116)( 70,120)( 71,125)( 72,124)
( 73,123)( 74,122)( 75,121)( 77, 80)( 78, 79)( 81, 82)( 83, 85)( 86, 88)
( 89, 90)( 91, 94)( 92, 93)( 96,100)( 97, 99)(126,276)(127,280)(128,279)
(129,278)(130,277)(131,282)(132,281)(133,285)(134,284)(135,283)(136,288)
(137,287)(138,286)(139,290)(140,289)(141,294)(142,293)(143,292)(144,291)
(145,295)(146,300)(147,299)(148,298)(149,297)(150,296)(151,251)(152,255)
(153,254)(154,253)(155,252)(156,257)(157,256)(158,260)(159,259)(160,258)
(161,263)(162,262)(163,261)(164,265)(165,264)(166,269)(167,268)(168,267)
(169,266)(170,270)(171,275)(172,274)(173,273)(174,272)(175,271)(176,351)
(177,355)(178,354)(179,353)(180,352)(181,357)(182,356)(183,360)(184,359)
(185,358)(186,363)(187,362)(188,361)(189,365)(190,364)(191,369)(192,368)
(193,367)(194,366)(195,370)(196,375)(197,374)(198,373)(199,372)(200,371)
(201,326)(202,330)(203,329)(204,328)(205,327)(206,332)(207,331)(208,335)
(209,334)(210,333)(211,338)(212,337)(213,336)(214,340)(215,339)(216,344)
(217,343)(218,342)(219,341)(220,345)(221,350)(222,349)(223,348)(224,347)
(225,346)(226,301)(227,305)(228,304)(229,303)(230,302)(231,307)(232,306)
(233,310)(234,309)(235,308)(236,313)(237,312)(238,311)(239,315)(240,314)
(241,319)(242,318)(243,317)(244,316)(245,320)(246,325)(247,324)(248,323)
(249,322)(250,321);
s3 := Sym(375)!( 1,126)( 2,130)( 3,129)( 4,128)( 5,127)( 6,131)( 7,135)
( 8,134)( 9,133)( 10,132)( 11,136)( 12,140)( 13,139)( 14,138)( 15,137)
( 16,141)( 17,145)( 18,144)( 19,143)( 20,142)( 21,146)( 22,150)( 23,149)
( 24,148)( 25,147)( 26,226)( 27,230)( 28,229)( 29,228)( 30,227)( 31,231)
( 32,235)( 33,234)( 34,233)( 35,232)( 36,236)( 37,240)( 38,239)( 39,238)
( 40,237)( 41,241)( 42,245)( 43,244)( 44,243)( 45,242)( 46,246)( 47,250)
( 48,249)( 49,248)( 50,247)( 51,201)( 52,205)( 53,204)( 54,203)( 55,202)
( 56,206)( 57,210)( 58,209)( 59,208)( 60,207)( 61,211)( 62,215)( 63,214)
( 64,213)( 65,212)( 66,216)( 67,220)( 68,219)( 69,218)( 70,217)( 71,221)
( 72,225)( 73,224)( 74,223)( 75,222)( 76,176)( 77,180)( 78,179)( 79,178)
( 80,177)( 81,181)( 82,185)( 83,184)( 84,183)( 85,182)( 86,186)( 87,190)
( 88,189)( 89,188)( 90,187)( 91,191)( 92,195)( 93,194)( 94,193)( 95,192)
( 96,196)( 97,200)( 98,199)( 99,198)(100,197)(101,151)(102,155)(103,154)
(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,161)
(112,165)(113,164)(114,163)(115,162)(116,166)(117,170)(118,169)(119,168)
(120,167)(121,171)(122,175)(123,174)(124,173)(125,172)(252,255)(253,254)
(257,260)(258,259)(262,265)(263,264)(267,270)(268,269)(272,275)(273,274)
(276,351)(277,355)(278,354)(279,353)(280,352)(281,356)(282,360)(283,359)
(284,358)(285,357)(286,361)(287,365)(288,364)(289,363)(290,362)(291,366)
(292,370)(293,369)(294,368)(295,367)(296,371)(297,375)(298,374)(299,373)
(300,372)(301,326)(302,330)(303,329)(304,328)(305,327)(306,331)(307,335)
(308,334)(309,333)(310,332)(311,336)(312,340)(313,339)(314,338)(315,337)
(316,341)(317,345)(318,344)(319,343)(320,342)(321,346)(322,350)(323,349)
(324,348)(325,347);
poly := sub<Sym(375)|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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope