Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,4,15}

Atlas Canonical Name {6,4,15}*1440

Overview

Group
SmallGroup(1440,5900)
Rank
4
Schläfli Type
{6,4,15}
Vertices, edges, …
6, 24, 60, 30
Order of s0s1s2s3
30
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

4-fold

5-fold

6-fold

8-fold

12-fold

15-fold

20-fold

24-fold

30-fold

36-fold

40-fold

60-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2)^2> of order 2

20 facets

6 vertex figures

Representations

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

References

None.

to this polytope.