Polytope of Type {10,72}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,72}*1440
Also Known As : {10,72|2}. if this polytope has another name.
Group : SmallGroup(1440,327)
Rank : 3
Schlafli Type : {10,72}
Number of vertices, edges, etc : 10, 360, 72
Order of s0s1s2 : 360
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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) :
2-fold quotients : {10,36}*720
3-fold quotients : {10,24}*480
4-fold quotients : {10,18}*360
5-fold quotients : {2,72}*288
6-fold quotients : {10,12}*240
9-fold quotients : {10,8}*160
10-fold quotients : {2,36}*144
12-fold quotients : {10,6}*120
15-fold quotients : {2,24}*96
18-fold quotients : {10,4}*80
20-fold quotients : {2,18}*72
30-fold quotients : {2,12}*48
36-fold quotients : {10,2}*40
40-fold quotients : {2,9}*36
45-fold quotients : {2,8}*32
60-fold quotients : {2,6}*24
72-fold quotients : {5,2}*20
90-fold quotients : {2,4}*16
120-fold quotients : {2,3}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 13)( 5, 14)( 6, 15)( 7, 10)( 8, 11)( 9, 12)( 19, 28)( 20, 29)
( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)( 37, 40)
( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)( 54, 57)
( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)( 80, 89)
( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)( 97,100)
( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)
(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)(140,149)
(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)(157,160)
(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177)
(184,193)(185,194)(186,195)(187,190)(188,191)(189,192)(199,208)(200,209)
(201,210)(202,205)(203,206)(204,207)(214,223)(215,224)(216,225)(217,220)
(218,221)(219,222)(229,238)(230,239)(231,240)(232,235)(233,236)(234,237)
(244,253)(245,254)(246,255)(247,250)(248,251)(249,252)(259,268)(260,269)
(261,270)(262,265)(263,266)(264,267)(274,283)(275,284)(276,285)(277,280)
(278,281)(279,282)(289,298)(290,299)(291,300)(292,295)(293,296)(294,297)
(304,313)(305,314)(306,315)(307,310)(308,311)(309,312)(319,328)(320,329)
(321,330)(322,325)(323,326)(324,327)(334,343)(335,344)(336,345)(337,340)
(338,341)(339,342)(349,358)(350,359)(351,360)(352,355)(353,356)(354,357);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 7, 13)( 8, 15)( 9, 14)( 11, 12)( 16, 36)
( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 46, 49)( 47, 51)
( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 81)( 62, 80)( 63, 79)
( 64, 78)( 65, 77)( 66, 76)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 86)
( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 91,139)( 92,141)( 93,140)( 94,136)
( 95,138)( 96,137)( 97,148)( 98,150)( 99,149)(100,145)(101,147)(102,146)
(103,142)(104,144)(105,143)(106,171)(107,170)(108,169)(109,168)(110,167)
(111,166)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)
(119,173)(120,172)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)
(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)
(135,157)(181,274)(182,276)(183,275)(184,271)(185,273)(186,272)(187,283)
(188,285)(189,284)(190,280)(191,282)(192,281)(193,277)(194,279)(195,278)
(196,306)(197,305)(198,304)(199,303)(200,302)(201,301)(202,315)(203,314)
(204,313)(205,312)(206,311)(207,310)(208,309)(209,308)(210,307)(211,291)
(212,290)(213,289)(214,288)(215,287)(216,286)(217,300)(218,299)(219,298)
(220,297)(221,296)(222,295)(223,294)(224,293)(225,292)(226,319)(227,321)
(228,320)(229,316)(230,318)(231,317)(232,328)(233,330)(234,329)(235,325)
(236,327)(237,326)(238,322)(239,324)(240,323)(241,351)(242,350)(243,349)
(244,348)(245,347)(246,346)(247,360)(248,359)(249,358)(250,357)(251,356)
(252,355)(253,354)(254,353)(255,352)(256,336)(257,335)(258,334)(259,333)
(260,332)(261,331)(262,345)(263,344)(264,343)(265,342)(266,341)(267,340)
(268,339)(269,338)(270,337);;
s2 := ( 1,196)( 2,198)( 3,197)( 4,199)( 5,201)( 6,200)( 7,202)( 8,204)
( 9,203)( 10,205)( 11,207)( 12,206)( 13,208)( 14,210)( 15,209)( 16,181)
( 17,183)( 18,182)( 19,184)( 20,186)( 21,185)( 22,187)( 23,189)( 24,188)
( 25,190)( 26,192)( 27,191)( 28,193)( 29,195)( 30,194)( 31,213)( 32,212)
( 33,211)( 34,216)( 35,215)( 36,214)( 37,219)( 38,218)( 39,217)( 40,222)
( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,241)( 47,243)( 48,242)
( 49,244)( 50,246)( 51,245)( 52,247)( 53,249)( 54,248)( 55,250)( 56,252)
( 57,251)( 58,253)( 59,255)( 60,254)( 61,226)( 62,228)( 63,227)( 64,229)
( 65,231)( 66,230)( 67,232)( 68,234)( 69,233)( 70,235)( 71,237)( 72,236)
( 73,238)( 74,240)( 75,239)( 76,258)( 77,257)( 78,256)( 79,261)( 80,260)
( 81,259)( 82,264)( 83,263)( 84,262)( 85,267)( 86,266)( 87,265)( 88,270)
( 89,269)( 90,268)( 91,331)( 92,333)( 93,332)( 94,334)( 95,336)( 96,335)
( 97,337)( 98,339)( 99,338)(100,340)(101,342)(102,341)(103,343)(104,345)
(105,344)(106,316)(107,318)(108,317)(109,319)(110,321)(111,320)(112,322)
(113,324)(114,323)(115,325)(116,327)(117,326)(118,328)(119,330)(120,329)
(121,348)(122,347)(123,346)(124,351)(125,350)(126,349)(127,354)(128,353)
(129,352)(130,357)(131,356)(132,355)(133,360)(134,359)(135,358)(136,286)
(137,288)(138,287)(139,289)(140,291)(141,290)(142,292)(143,294)(144,293)
(145,295)(146,297)(147,296)(148,298)(149,300)(150,299)(151,271)(152,273)
(153,272)(154,274)(155,276)(156,275)(157,277)(158,279)(159,278)(160,280)
(161,282)(162,281)(163,283)(164,285)(165,284)(166,303)(167,302)(168,301)
(169,306)(170,305)(171,304)(172,309)(173,308)(174,307)(175,312)(176,311)
(177,310)(178,315)(179,314)(180,313);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*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,
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*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*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*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)!( 4, 13)( 5, 14)( 6, 15)( 7, 10)( 8, 11)( 9, 12)( 19, 28)
( 20, 29)( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)
( 37, 40)( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)
( 54, 57)( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)
( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)
( 97,100)( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)
(114,117)(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)
(140,149)(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)
(157,160)(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)
(174,177)(184,193)(185,194)(186,195)(187,190)(188,191)(189,192)(199,208)
(200,209)(201,210)(202,205)(203,206)(204,207)(214,223)(215,224)(216,225)
(217,220)(218,221)(219,222)(229,238)(230,239)(231,240)(232,235)(233,236)
(234,237)(244,253)(245,254)(246,255)(247,250)(248,251)(249,252)(259,268)
(260,269)(261,270)(262,265)(263,266)(264,267)(274,283)(275,284)(276,285)
(277,280)(278,281)(279,282)(289,298)(290,299)(291,300)(292,295)(293,296)
(294,297)(304,313)(305,314)(306,315)(307,310)(308,311)(309,312)(319,328)
(320,329)(321,330)(322,325)(323,326)(324,327)(334,343)(335,344)(336,345)
(337,340)(338,341)(339,342)(349,358)(350,359)(351,360)(352,355)(353,356)
(354,357);
s1 := Sym(360)!( 1, 4)( 2, 6)( 3, 5)( 7, 13)( 8, 15)( 9, 14)( 11, 12)
( 16, 36)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 46, 49)
( 47, 51)( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 81)( 62, 80)
( 63, 79)( 64, 78)( 65, 77)( 66, 76)( 67, 90)( 68, 89)( 69, 88)( 70, 87)
( 71, 86)( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 91,139)( 92,141)( 93,140)
( 94,136)( 95,138)( 96,137)( 97,148)( 98,150)( 99,149)(100,145)(101,147)
(102,146)(103,142)(104,144)(105,143)(106,171)(107,170)(108,169)(109,168)
(110,167)(111,166)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)
(118,174)(119,173)(120,172)(121,156)(122,155)(123,154)(124,153)(125,152)
(126,151)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)
(134,158)(135,157)(181,274)(182,276)(183,275)(184,271)(185,273)(186,272)
(187,283)(188,285)(189,284)(190,280)(191,282)(192,281)(193,277)(194,279)
(195,278)(196,306)(197,305)(198,304)(199,303)(200,302)(201,301)(202,315)
(203,314)(204,313)(205,312)(206,311)(207,310)(208,309)(209,308)(210,307)
(211,291)(212,290)(213,289)(214,288)(215,287)(216,286)(217,300)(218,299)
(219,298)(220,297)(221,296)(222,295)(223,294)(224,293)(225,292)(226,319)
(227,321)(228,320)(229,316)(230,318)(231,317)(232,328)(233,330)(234,329)
(235,325)(236,327)(237,326)(238,322)(239,324)(240,323)(241,351)(242,350)
(243,349)(244,348)(245,347)(246,346)(247,360)(248,359)(249,358)(250,357)
(251,356)(252,355)(253,354)(254,353)(255,352)(256,336)(257,335)(258,334)
(259,333)(260,332)(261,331)(262,345)(263,344)(264,343)(265,342)(266,341)
(267,340)(268,339)(269,338)(270,337);
s2 := Sym(360)!( 1,196)( 2,198)( 3,197)( 4,199)( 5,201)( 6,200)( 7,202)
( 8,204)( 9,203)( 10,205)( 11,207)( 12,206)( 13,208)( 14,210)( 15,209)
( 16,181)( 17,183)( 18,182)( 19,184)( 20,186)( 21,185)( 22,187)( 23,189)
( 24,188)( 25,190)( 26,192)( 27,191)( 28,193)( 29,195)( 30,194)( 31,213)
( 32,212)( 33,211)( 34,216)( 35,215)( 36,214)( 37,219)( 38,218)( 39,217)
( 40,222)( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,241)( 47,243)
( 48,242)( 49,244)( 50,246)( 51,245)( 52,247)( 53,249)( 54,248)( 55,250)
( 56,252)( 57,251)( 58,253)( 59,255)( 60,254)( 61,226)( 62,228)( 63,227)
( 64,229)( 65,231)( 66,230)( 67,232)( 68,234)( 69,233)( 70,235)( 71,237)
( 72,236)( 73,238)( 74,240)( 75,239)( 76,258)( 77,257)( 78,256)( 79,261)
( 80,260)( 81,259)( 82,264)( 83,263)( 84,262)( 85,267)( 86,266)( 87,265)
( 88,270)( 89,269)( 90,268)( 91,331)( 92,333)( 93,332)( 94,334)( 95,336)
( 96,335)( 97,337)( 98,339)( 99,338)(100,340)(101,342)(102,341)(103,343)
(104,345)(105,344)(106,316)(107,318)(108,317)(109,319)(110,321)(111,320)
(112,322)(113,324)(114,323)(115,325)(116,327)(117,326)(118,328)(119,330)
(120,329)(121,348)(122,347)(123,346)(124,351)(125,350)(126,349)(127,354)
(128,353)(129,352)(130,357)(131,356)(132,355)(133,360)(134,359)(135,358)
(136,286)(137,288)(138,287)(139,289)(140,291)(141,290)(142,292)(143,294)
(144,293)(145,295)(146,297)(147,296)(148,298)(149,300)(150,299)(151,271)
(152,273)(153,272)(154,274)(155,276)(156,275)(157,277)(158,279)(159,278)
(160,280)(161,282)(162,281)(163,283)(164,285)(165,284)(166,303)(167,302)
(168,301)(169,306)(170,305)(171,304)(172,309)(173,308)(174,307)(175,312)
(176,311)(177,310)(178,315)(179,314)(180,313);
poly := sub<Sym(360)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*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,
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*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*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*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