Polytope of Type {72,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,10}*1440
Also Known As : {72,10|2}. if this polytope has another name.
Group : SmallGroup(1440,327)
Rank : 3
Schlafli Type : {72,10}
Number of vertices, edges, etc : 72, 360, 10
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 : {36,10}*720
   3-fold quotients : {24,10}*480
   4-fold quotients : {18,10}*360
   5-fold quotients : {72,2}*288
   6-fold quotients : {12,10}*240
   9-fold quotients : {8,10}*160
   10-fold quotients : {36,2}*144
   12-fold quotients : {6,10}*120
   15-fold quotients : {24,2}*96
   18-fold quotients : {4,10}*80
   20-fold quotients : {18,2}*72
   30-fold quotients : {12,2}*48
   36-fold quotients : {2,10}*40
   40-fold quotients : {9,2}*36
   45-fold quotients : {8,2}*32
   60-fold quotients : {6,2}*24
   72-fold quotients : {2,5}*20
   90-fold quotients : {4,2}*16
   120-fold quotients : {3,2}*12
   180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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