# Polytope of Type {4,45}

Atlas Canonical Name : {4,45}*720
if this polytope has a name.
Group : SmallGroup(720,397)
Rank : 3
Schlafli Type : {4,45}
Number of vertices, edges, etc : 8, 180, 90
Order of s0s1s2 : 90
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,45,2} of size 1440
Vertex Figure Of :
{2,4,45} of size 1440
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,45}*360
3-fold quotients : {4,15}*240
4-fold quotients : {2,45}*180
5-fold quotients : {4,9}*144
6-fold quotients : {4,15}*120
10-fold quotients : {4,9}*72
12-fold quotients : {2,15}*60
15-fold quotients : {4,3}*48
20-fold quotients : {2,9}*36
30-fold quotients : {4,3}*24
36-fold quotients : {2,5}*20
60-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,45}*1440, {4,90}*1440
Permutation Representation (GAP) :
```s0 := (  1,183)(  2,184)(  3,181)(  4,182)(  5,187)(  6,188)(  7,185)(  8,186)
(  9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)( 16,194)
( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)( 24,202)
( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)( 32,210)
( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)( 40,218)
( 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)( 61,243)( 62,244)( 63,241)( 64,242)
( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)( 72,250)
( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)( 80,258)
( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)( 88,266)
( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)( 96,274)
( 97,279)( 98,280)( 99,277)(100,278)(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)
(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)(128,306)
(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)(136,314)
(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)(144,322)
(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)(152,330)
(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)(160,338)
(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);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)
( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)
( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)
( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)
( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)
( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)
( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)
( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)
(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)
(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)
(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)(194,231)
(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)(202,235)
(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)(210,227)
(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)(242,307)
(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)(250,311)
(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)(258,351)
(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)(266,343)
(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)(274,347)
(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)(282,327)
(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)(290,319)
(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)(298,323)
(299,322)(300,324);;
s2 := (  1,133)(  2,136)(  3,135)(  4,134)(  5,141)(  6,144)(  7,143)(  8,142)
(  9,137)( 10,140)( 11,139)( 12,138)( 13,121)( 14,124)( 15,123)( 16,122)
( 17,129)( 18,132)( 19,131)( 20,130)( 21,125)( 22,128)( 23,127)( 24,126)
( 25,169)( 26,172)( 27,171)( 28,170)( 29,177)( 30,180)( 31,179)( 32,178)
( 33,173)( 34,176)( 35,175)( 36,174)( 37,157)( 38,160)( 39,159)( 40,158)
( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)
( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)
( 57,149)( 58,152)( 59,151)( 60,150)( 61, 73)( 62, 76)( 63, 75)( 64, 74)
( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)
( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)
( 93,113)( 94,116)( 95,115)( 96,114)( 98,100)(101,105)(102,108)(103,107)
(104,106)(181,313)(182,316)(183,315)(184,314)(185,321)(186,324)(187,323)
(188,322)(189,317)(190,320)(191,319)(192,318)(193,301)(194,304)(195,303)
(196,302)(197,309)(198,312)(199,311)(200,310)(201,305)(202,308)(203,307)
(204,306)(205,349)(206,352)(207,351)(208,350)(209,357)(210,360)(211,359)
(212,358)(213,353)(214,356)(215,355)(216,354)(217,337)(218,340)(219,339)
(220,338)(221,345)(222,348)(223,347)(224,346)(225,341)(226,344)(227,343)
(228,342)(229,325)(230,328)(231,327)(232,326)(233,333)(234,336)(235,335)
(236,334)(237,329)(238,332)(239,331)(240,330)(241,253)(242,256)(243,255)
(244,254)(245,261)(246,264)(247,263)(248,262)(249,257)(250,260)(251,259)
(252,258)(265,289)(266,292)(267,291)(268,290)(269,297)(270,300)(271,299)
(272,298)(273,293)(274,296)(275,295)(276,294)(278,280)(281,285)(282,288)
(283,287)(284,286);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope