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# Polytope of Type {2,90,4}

Atlas Canonical Name : {2,90,4}*1440c
if this polytope has a name.
Group : SmallGroup(1440,4575)
Rank : 4
Schlafli Type : {2,90,4}
Number of vertices, edges, etc : 2, 90, 180, 4
Order of s0s1s2s3 : 90
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,45,4}*720
3-fold quotients : {2,30,4}*480c
5-fold quotients : {2,18,4}*288c
6-fold quotients : {2,15,4}*240
10-fold quotients : {2,9,4}*144
15-fold quotients : {2,6,4}*96b
30-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 51)( 16, 53)( 17, 52)
( 18, 54)( 19, 59)( 20, 61)( 21, 60)( 22, 62)( 23, 55)( 24, 57)( 25, 56)
( 26, 58)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)( 33, 48)
( 34, 50)( 35, 43)( 36, 45)( 37, 44)( 38, 46)( 63,127)( 64,129)( 65,128)
( 66,130)( 67,123)( 68,125)( 69,124)( 70,126)( 71,131)( 72,133)( 73,132)
( 74,134)( 75,175)( 76,177)( 77,176)( 78,178)( 79,171)( 80,173)( 81,172)
( 82,174)( 83,179)( 84,181)( 85,180)( 86,182)( 87,163)( 88,165)( 89,164)
( 90,166)( 91,159)( 92,161)( 93,160)( 94,162)( 95,167)( 96,169)( 97,168)
( 98,170)( 99,151)(100,153)(101,152)(102,154)(103,147)(104,149)(105,148)
(106,150)(107,155)(108,157)(109,156)(110,158)(111,139)(112,141)(113,140)
(114,142)(115,135)(116,137)(117,136)(118,138)(119,143)(120,145)(121,144)
(122,146)(184,185)(187,191)(188,193)(189,192)(190,194)(195,231)(196,233)
(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)(204,237)
(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)(212,229)
(213,228)(214,230)(215,223)(216,225)(217,224)(218,226)(243,307)(244,309)
(245,308)(246,310)(247,303)(248,305)(249,304)(250,306)(251,311)(252,313)
(253,312)(254,314)(255,355)(256,357)(257,356)(258,358)(259,351)(260,353)
(261,352)(262,354)(263,359)(264,361)(265,360)(266,362)(267,343)(268,345)
(269,344)(270,346)(271,339)(272,341)(273,340)(274,342)(275,347)(276,349)
(277,348)(278,350)(279,331)(280,333)(281,332)(282,334)(283,327)(284,329)
(285,328)(286,330)(287,335)(288,337)(289,336)(290,338)(291,319)(292,321)
(293,320)(294,322)(295,315)(296,317)(297,316)(298,318)(299,323)(300,325)
(301,324)(302,326);;
s2 := (  3,255)(  4,256)(  5,258)(  6,257)(  7,263)(  8,264)(  9,266)( 10,265)
( 11,259)( 12,260)( 13,262)( 14,261)( 15,243)( 16,244)( 17,246)( 18,245)
( 19,251)( 20,252)( 21,254)( 22,253)( 23,247)( 24,248)( 25,250)( 26,249)
( 27,291)( 28,292)( 29,294)( 30,293)( 31,299)( 32,300)( 33,302)( 34,301)
( 35,295)( 36,296)( 37,298)( 38,297)( 39,279)( 40,280)( 41,282)( 42,281)
( 43,287)( 44,288)( 45,290)( 46,289)( 47,283)( 48,284)( 49,286)( 50,285)
( 51,267)( 52,268)( 53,270)( 54,269)( 55,275)( 56,276)( 57,278)( 58,277)
( 59,271)( 60,272)( 61,274)( 62,273)( 63,195)( 64,196)( 65,198)( 66,197)
( 67,203)( 68,204)( 69,206)( 70,205)( 71,199)( 72,200)( 73,202)( 74,201)
( 75,183)( 76,184)( 77,186)( 78,185)( 79,191)( 80,192)( 81,194)( 82,193)
( 83,187)( 84,188)( 85,190)( 86,189)( 87,231)( 88,232)( 89,234)( 90,233)
( 91,239)( 92,240)( 93,242)( 94,241)( 95,235)( 96,236)( 97,238)( 98,237)
( 99,219)(100,220)(101,222)(102,221)(103,227)(104,228)(105,230)(106,229)
(107,223)(108,224)(109,226)(110,225)(111,207)(112,208)(113,210)(114,209)
(115,215)(116,216)(117,218)(118,217)(119,211)(120,212)(121,214)(122,213)
(123,319)(124,320)(125,322)(126,321)(127,315)(128,316)(129,318)(130,317)
(131,323)(132,324)(133,326)(134,325)(135,307)(136,308)(137,310)(138,309)
(139,303)(140,304)(141,306)(142,305)(143,311)(144,312)(145,314)(146,313)
(147,355)(148,356)(149,358)(150,357)(151,351)(152,352)(153,354)(154,353)
(155,359)(156,360)(157,362)(158,361)(159,343)(160,344)(161,346)(162,345)
(163,339)(164,340)(165,342)(166,341)(167,347)(168,348)(169,350)(170,349)
(171,331)(172,332)(173,334)(174,333)(175,327)(176,328)(177,330)(178,329)
(179,335)(180,336)(181,338)(182,337);;
s3 := (  3,186)(  4,185)(  5,184)(  6,183)(  7,190)(  8,189)(  9,188)( 10,187)
( 11,194)( 12,193)( 13,192)( 14,191)( 15,198)( 16,197)( 17,196)( 18,195)
( 19,202)( 20,201)( 21,200)( 22,199)( 23,206)( 24,205)( 25,204)( 26,203)
( 27,210)( 28,209)( 29,208)( 30,207)( 31,214)( 32,213)( 33,212)( 34,211)
( 35,218)( 36,217)( 37,216)( 38,215)( 39,222)( 40,221)( 41,220)( 42,219)
( 43,226)( 44,225)( 45,224)( 46,223)( 47,230)( 48,229)( 49,228)( 50,227)
( 51,234)( 52,233)( 53,232)( 54,231)( 55,238)( 56,237)( 57,236)( 58,235)
( 59,242)( 60,241)( 61,240)( 62,239)( 63,246)( 64,245)( 65,244)( 66,243)
( 67,250)( 68,249)( 69,248)( 70,247)( 71,254)( 72,253)( 73,252)( 74,251)
( 75,258)( 76,257)( 77,256)( 78,255)( 79,262)( 80,261)( 81,260)( 82,259)
( 83,266)( 84,265)( 85,264)( 86,263)( 87,270)( 88,269)( 89,268)( 90,267)
( 91,274)( 92,273)( 93,272)( 94,271)( 95,278)( 96,277)( 97,276)( 98,275)
( 99,282)(100,281)(101,280)(102,279)(103,286)(104,285)(105,284)(106,283)
(107,290)(108,289)(109,288)(110,287)(111,294)(112,293)(113,292)(114,291)
(115,298)(116,297)(117,296)(118,295)(119,302)(120,301)(121,300)(122,299)
(123,306)(124,305)(125,304)(126,303)(127,310)(128,309)(129,308)(130,307)
(131,314)(132,313)(133,312)(134,311)(135,318)(136,317)(137,316)(138,315)
(139,322)(140,321)(141,320)(142,319)(143,326)(144,325)(145,324)(146,323)
(147,330)(148,329)(149,328)(150,327)(151,334)(152,333)(153,332)(154,331)
(155,338)(156,337)(157,336)(158,335)(159,342)(160,341)(161,340)(162,339)
(163,346)(164,345)(165,344)(166,343)(167,350)(168,349)(169,348)(170,347)
(171,354)(172,353)(173,352)(174,351)(175,358)(176,357)(177,356)(178,355)
(179,362)(180,361)(181,360)(182,359);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*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(362)!(1,2);
s1 := Sym(362)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 51)( 16, 53)
( 17, 52)( 18, 54)( 19, 59)( 20, 61)( 21, 60)( 22, 62)( 23, 55)( 24, 57)
( 25, 56)( 26, 58)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)
( 33, 48)( 34, 50)( 35, 43)( 36, 45)( 37, 44)( 38, 46)( 63,127)( 64,129)
( 65,128)( 66,130)( 67,123)( 68,125)( 69,124)( 70,126)( 71,131)( 72,133)
( 73,132)( 74,134)( 75,175)( 76,177)( 77,176)( 78,178)( 79,171)( 80,173)
( 81,172)( 82,174)( 83,179)( 84,181)( 85,180)( 86,182)( 87,163)( 88,165)
( 89,164)( 90,166)( 91,159)( 92,161)( 93,160)( 94,162)( 95,167)( 96,169)
( 97,168)( 98,170)( 99,151)(100,153)(101,152)(102,154)(103,147)(104,149)
(105,148)(106,150)(107,155)(108,157)(109,156)(110,158)(111,139)(112,141)
(113,140)(114,142)(115,135)(116,137)(117,136)(118,138)(119,143)(120,145)
(121,144)(122,146)(184,185)(187,191)(188,193)(189,192)(190,194)(195,231)
(196,233)(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)
(204,237)(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)
(212,229)(213,228)(214,230)(215,223)(216,225)(217,224)(218,226)(243,307)
(244,309)(245,308)(246,310)(247,303)(248,305)(249,304)(250,306)(251,311)
(252,313)(253,312)(254,314)(255,355)(256,357)(257,356)(258,358)(259,351)
(260,353)(261,352)(262,354)(263,359)(264,361)(265,360)(266,362)(267,343)
(268,345)(269,344)(270,346)(271,339)(272,341)(273,340)(274,342)(275,347)
(276,349)(277,348)(278,350)(279,331)(280,333)(281,332)(282,334)(283,327)
(284,329)(285,328)(286,330)(287,335)(288,337)(289,336)(290,338)(291,319)
(292,321)(293,320)(294,322)(295,315)(296,317)(297,316)(298,318)(299,323)
(300,325)(301,324)(302,326);
s2 := Sym(362)!(  3,255)(  4,256)(  5,258)(  6,257)(  7,263)(  8,264)(  9,266)
( 10,265)( 11,259)( 12,260)( 13,262)( 14,261)( 15,243)( 16,244)( 17,246)
( 18,245)( 19,251)( 20,252)( 21,254)( 22,253)( 23,247)( 24,248)( 25,250)
( 26,249)( 27,291)( 28,292)( 29,294)( 30,293)( 31,299)( 32,300)( 33,302)
( 34,301)( 35,295)( 36,296)( 37,298)( 38,297)( 39,279)( 40,280)( 41,282)
( 42,281)( 43,287)( 44,288)( 45,290)( 46,289)( 47,283)( 48,284)( 49,286)
( 50,285)( 51,267)( 52,268)( 53,270)( 54,269)( 55,275)( 56,276)( 57,278)
( 58,277)( 59,271)( 60,272)( 61,274)( 62,273)( 63,195)( 64,196)( 65,198)
( 66,197)( 67,203)( 68,204)( 69,206)( 70,205)( 71,199)( 72,200)( 73,202)
( 74,201)( 75,183)( 76,184)( 77,186)( 78,185)( 79,191)( 80,192)( 81,194)
( 82,193)( 83,187)( 84,188)( 85,190)( 86,189)( 87,231)( 88,232)( 89,234)
( 90,233)( 91,239)( 92,240)( 93,242)( 94,241)( 95,235)( 96,236)( 97,238)
( 98,237)( 99,219)(100,220)(101,222)(102,221)(103,227)(104,228)(105,230)
(106,229)(107,223)(108,224)(109,226)(110,225)(111,207)(112,208)(113,210)
(114,209)(115,215)(116,216)(117,218)(118,217)(119,211)(120,212)(121,214)
(122,213)(123,319)(124,320)(125,322)(126,321)(127,315)(128,316)(129,318)
(130,317)(131,323)(132,324)(133,326)(134,325)(135,307)(136,308)(137,310)
(138,309)(139,303)(140,304)(141,306)(142,305)(143,311)(144,312)(145,314)
(146,313)(147,355)(148,356)(149,358)(150,357)(151,351)(152,352)(153,354)
(154,353)(155,359)(156,360)(157,362)(158,361)(159,343)(160,344)(161,346)
(162,345)(163,339)(164,340)(165,342)(166,341)(167,347)(168,348)(169,350)
(170,349)(171,331)(172,332)(173,334)(174,333)(175,327)(176,328)(177,330)
(178,329)(179,335)(180,336)(181,338)(182,337);
s3 := Sym(362)!(  3,186)(  4,185)(  5,184)(  6,183)(  7,190)(  8,189)(  9,188)
( 10,187)( 11,194)( 12,193)( 13,192)( 14,191)( 15,198)( 16,197)( 17,196)
( 18,195)( 19,202)( 20,201)( 21,200)( 22,199)( 23,206)( 24,205)( 25,204)
( 26,203)( 27,210)( 28,209)( 29,208)( 30,207)( 31,214)( 32,213)( 33,212)
( 34,211)( 35,218)( 36,217)( 37,216)( 38,215)( 39,222)( 40,221)( 41,220)
( 42,219)( 43,226)( 44,225)( 45,224)( 46,223)( 47,230)( 48,229)( 49,228)
( 50,227)( 51,234)( 52,233)( 53,232)( 54,231)( 55,238)( 56,237)( 57,236)
( 58,235)( 59,242)( 60,241)( 61,240)( 62,239)( 63,246)( 64,245)( 65,244)
( 66,243)( 67,250)( 68,249)( 69,248)( 70,247)( 71,254)( 72,253)( 73,252)
( 74,251)( 75,258)( 76,257)( 77,256)( 78,255)( 79,262)( 80,261)( 81,260)
( 82,259)( 83,266)( 84,265)( 85,264)( 86,263)( 87,270)( 88,269)( 89,268)
( 90,267)( 91,274)( 92,273)( 93,272)( 94,271)( 95,278)( 96,277)( 97,276)
( 98,275)( 99,282)(100,281)(101,280)(102,279)(103,286)(104,285)(105,284)
(106,283)(107,290)(108,289)(109,288)(110,287)(111,294)(112,293)(113,292)
(114,291)(115,298)(116,297)(117,296)(118,295)(119,302)(120,301)(121,300)
(122,299)(123,306)(124,305)(125,304)(126,303)(127,310)(128,309)(129,308)
(130,307)(131,314)(132,313)(133,312)(134,311)(135,318)(136,317)(137,316)
(138,315)(139,322)(140,321)(141,320)(142,319)(143,326)(144,325)(145,324)
(146,323)(147,330)(148,329)(149,328)(150,327)(151,334)(152,333)(153,332)
(154,331)(155,338)(156,337)(157,336)(158,335)(159,342)(160,341)(161,340)
(162,339)(163,346)(164,345)(165,344)(166,343)(167,350)(168,349)(169,348)
(170,347)(171,354)(172,353)(173,352)(174,351)(175,358)(176,357)(177,356)
(178,355)(179,362)(180,361)(181,360)(182,359);
poly := sub<Sym(362)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*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 >;

```

to this polytope