# Polytope of Type {4,200}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,200}*1600a
Also Known As : {4,200|2}. if this polytope has another name.
Group : SmallGroup(1600,361)
Rank : 3
Schlafli Type : {4,200}
Number of vertices, edges, etc : 4, 400, 200
Order of s0s1s2 : 200
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 : {4,100}*800, {2,200}*800
4-fold quotients : {2,100}*400, {4,50}*400
5-fold quotients : {4,40}*320a
8-fold quotients : {2,50}*200
10-fold quotients : {4,20}*160, {2,40}*160
16-fold quotients : {2,25}*100
20-fold quotients : {2,20}*80, {4,10}*80
25-fold quotients : {4,8}*64a
40-fold quotients : {2,10}*40
50-fold quotients : {4,4}*32, {2,8}*32
80-fold quotients : {2,5}*20
100-fold quotients : {2,4}*16, {4,2}*16
200-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (201,301)(202,302)(203,303)(204,304)(205,305)(206,306)(207,307)(208,308)
(209,309)(210,310)(211,311)(212,312)(213,313)(214,314)(215,315)(216,316)
(217,317)(218,318)(219,319)(220,320)(221,321)(222,322)(223,323)(224,324)
(225,325)(226,326)(227,327)(228,328)(229,329)(230,330)(231,331)(232,332)
(233,333)(234,334)(235,335)(236,336)(237,337)(238,338)(239,339)(240,340)
(241,341)(242,342)(243,343)(244,344)(245,345)(246,346)(247,347)(248,348)
(249,349)(250,350)(251,351)(252,352)(253,353)(254,354)(255,355)(256,356)
(257,357)(258,358)(259,359)(260,360)(261,361)(262,362)(263,363)(264,364)
(265,365)(266,366)(267,367)(268,368)(269,369)(270,370)(271,371)(272,372)
(273,373)(274,374)(275,375)(276,376)(277,377)(278,378)(279,379)(280,380)
(281,381)(282,382)(283,383)(284,384)(285,385)(286,386)(287,387)(288,388)
(289,389)(290,390)(291,391)(292,392)(293,393)(294,394)(295,395)(296,396)
(297,397)(298,398)(299,399)(300,400);;
s1 := (  1,201)(  2,205)(  3,204)(  4,203)(  5,202)(  6,225)(  7,224)(  8,223)
(  9,222)( 10,221)( 11,220)( 12,219)( 13,218)( 14,217)( 15,216)( 16,215)
( 17,214)( 18,213)( 19,212)( 20,211)( 21,210)( 22,209)( 23,208)( 24,207)
( 25,206)( 26,226)( 27,230)( 28,229)( 29,228)( 30,227)( 31,250)( 32,249)
( 33,248)( 34,247)( 35,246)( 36,245)( 37,244)( 38,243)( 39,242)( 40,241)
( 41,240)( 42,239)( 43,238)( 44,237)( 45,236)( 46,235)( 47,234)( 48,233)
( 49,232)( 50,231)( 51,276)( 52,280)( 53,279)( 54,278)( 55,277)( 56,300)
( 57,299)( 58,298)( 59,297)( 60,296)( 61,295)( 62,294)( 63,293)( 64,292)
( 65,291)( 66,290)( 67,289)( 68,288)( 69,287)( 70,286)( 71,285)( 72,284)
( 73,283)( 74,282)( 75,281)( 76,251)( 77,255)( 78,254)( 79,253)( 80,252)
( 81,275)( 82,274)( 83,273)( 84,272)( 85,271)( 86,270)( 87,269)( 88,268)
( 89,267)( 90,266)( 91,265)( 92,264)( 93,263)( 94,262)( 95,261)( 96,260)
( 97,259)( 98,258)( 99,257)(100,256)(101,301)(102,305)(103,304)(104,303)
(105,302)(106,325)(107,324)(108,323)(109,322)(110,321)(111,320)(112,319)
(113,318)(114,317)(115,316)(116,315)(117,314)(118,313)(119,312)(120,311)
(121,310)(122,309)(123,308)(124,307)(125,306)(126,326)(127,330)(128,329)
(129,328)(130,327)(131,350)(132,349)(133,348)(134,347)(135,346)(136,345)
(137,344)(138,343)(139,342)(140,341)(141,340)(142,339)(143,338)(144,337)
(145,336)(146,335)(147,334)(148,333)(149,332)(150,331)(151,376)(152,380)
(153,379)(154,378)(155,377)(156,400)(157,399)(158,398)(159,397)(160,396)
(161,395)(162,394)(163,393)(164,392)(165,391)(166,390)(167,389)(168,388)
(169,387)(170,386)(171,385)(172,384)(173,383)(174,382)(175,381)(176,351)
(177,355)(178,354)(179,353)(180,352)(181,375)(182,374)(183,373)(184,372)
(185,371)(186,370)(187,369)(188,368)(189,367)(190,366)(191,365)(192,364)
(193,363)(194,362)(195,361)(196,360)(197,359)(198,358)(199,357)(200,356);;
s2 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 25)( 12, 24)( 13, 23)
( 14, 22)( 15, 21)( 16, 20)( 17, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)
( 30, 32)( 36, 50)( 37, 49)( 38, 48)( 39, 47)( 40, 46)( 41, 45)( 42, 44)
( 51, 81)( 52, 85)( 53, 84)( 54, 83)( 55, 82)( 56, 76)( 57, 80)( 58, 79)
( 59, 78)( 60, 77)( 61,100)( 62, 99)( 63, 98)( 64, 97)( 65, 96)( 66, 95)
( 67, 94)( 68, 93)( 69, 92)( 70, 91)( 71, 90)( 72, 89)( 73, 88)( 74, 87)
( 75, 86)(101,106)(102,110)(103,109)(104,108)(105,107)(111,125)(112,124)
(113,123)(114,122)(115,121)(116,120)(117,119)(126,131)(127,135)(128,134)
(129,133)(130,132)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)
(142,144)(151,181)(152,185)(153,184)(154,183)(155,182)(156,176)(157,180)
(158,179)(159,178)(160,177)(161,200)(162,199)(163,198)(164,197)(165,196)
(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)(173,188)
(174,187)(175,186)(201,256)(202,260)(203,259)(204,258)(205,257)(206,251)
(207,255)(208,254)(209,253)(210,252)(211,275)(212,274)(213,273)(214,272)
(215,271)(216,270)(217,269)(218,268)(219,267)(220,266)(221,265)(222,264)
(223,263)(224,262)(225,261)(226,281)(227,285)(228,284)(229,283)(230,282)
(231,276)(232,280)(233,279)(234,278)(235,277)(236,300)(237,299)(238,298)
(239,297)(240,296)(241,295)(242,294)(243,293)(244,292)(245,291)(246,290)
(247,289)(248,288)(249,287)(250,286)(301,356)(302,360)(303,359)(304,358)
(305,357)(306,351)(307,355)(308,354)(309,353)(310,352)(311,375)(312,374)
(313,373)(314,372)(315,371)(316,370)(317,369)(318,368)(319,367)(320,366)
(321,365)(322,364)(323,363)(324,362)(325,361)(326,381)(327,385)(328,384)
(329,383)(330,382)(331,376)(332,380)(333,379)(334,378)(335,377)(336,400)
(337,399)(338,398)(339,397)(340,396)(341,395)(342,394)(343,393)(344,392)
(345,391)(346,390)(347,389)(348,388)(349,387)(350,386);;
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*s0*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*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*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*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(400)!(201,301)(202,302)(203,303)(204,304)(205,305)(206,306)(207,307)
(208,308)(209,309)(210,310)(211,311)(212,312)(213,313)(214,314)(215,315)
(216,316)(217,317)(218,318)(219,319)(220,320)(221,321)(222,322)(223,323)
(224,324)(225,325)(226,326)(227,327)(228,328)(229,329)(230,330)(231,331)
(232,332)(233,333)(234,334)(235,335)(236,336)(237,337)(238,338)(239,339)
(240,340)(241,341)(242,342)(243,343)(244,344)(245,345)(246,346)(247,347)
(248,348)(249,349)(250,350)(251,351)(252,352)(253,353)(254,354)(255,355)
(256,356)(257,357)(258,358)(259,359)(260,360)(261,361)(262,362)(263,363)
(264,364)(265,365)(266,366)(267,367)(268,368)(269,369)(270,370)(271,371)
(272,372)(273,373)(274,374)(275,375)(276,376)(277,377)(278,378)(279,379)
(280,380)(281,381)(282,382)(283,383)(284,384)(285,385)(286,386)(287,387)
(288,388)(289,389)(290,390)(291,391)(292,392)(293,393)(294,394)(295,395)
(296,396)(297,397)(298,398)(299,399)(300,400);
s1 := Sym(400)!(  1,201)(  2,205)(  3,204)(  4,203)(  5,202)(  6,225)(  7,224)
(  8,223)(  9,222)( 10,221)( 11,220)( 12,219)( 13,218)( 14,217)( 15,216)
( 16,215)( 17,214)( 18,213)( 19,212)( 20,211)( 21,210)( 22,209)( 23,208)
( 24,207)( 25,206)( 26,226)( 27,230)( 28,229)( 29,228)( 30,227)( 31,250)
( 32,249)( 33,248)( 34,247)( 35,246)( 36,245)( 37,244)( 38,243)( 39,242)
( 40,241)( 41,240)( 42,239)( 43,238)( 44,237)( 45,236)( 46,235)( 47,234)
( 48,233)( 49,232)( 50,231)( 51,276)( 52,280)( 53,279)( 54,278)( 55,277)
( 56,300)( 57,299)( 58,298)( 59,297)( 60,296)( 61,295)( 62,294)( 63,293)
( 64,292)( 65,291)( 66,290)( 67,289)( 68,288)( 69,287)( 70,286)( 71,285)
( 72,284)( 73,283)( 74,282)( 75,281)( 76,251)( 77,255)( 78,254)( 79,253)
( 80,252)( 81,275)( 82,274)( 83,273)( 84,272)( 85,271)( 86,270)( 87,269)
( 88,268)( 89,267)( 90,266)( 91,265)( 92,264)( 93,263)( 94,262)( 95,261)
( 96,260)( 97,259)( 98,258)( 99,257)(100,256)(101,301)(102,305)(103,304)
(104,303)(105,302)(106,325)(107,324)(108,323)(109,322)(110,321)(111,320)
(112,319)(113,318)(114,317)(115,316)(116,315)(117,314)(118,313)(119,312)
(120,311)(121,310)(122,309)(123,308)(124,307)(125,306)(126,326)(127,330)
(128,329)(129,328)(130,327)(131,350)(132,349)(133,348)(134,347)(135,346)
(136,345)(137,344)(138,343)(139,342)(140,341)(141,340)(142,339)(143,338)
(144,337)(145,336)(146,335)(147,334)(148,333)(149,332)(150,331)(151,376)
(152,380)(153,379)(154,378)(155,377)(156,400)(157,399)(158,398)(159,397)
(160,396)(161,395)(162,394)(163,393)(164,392)(165,391)(166,390)(167,389)
(168,388)(169,387)(170,386)(171,385)(172,384)(173,383)(174,382)(175,381)
(176,351)(177,355)(178,354)(179,353)(180,352)(181,375)(182,374)(183,373)
(184,372)(185,371)(186,370)(187,369)(188,368)(189,367)(190,366)(191,365)
(192,364)(193,363)(194,362)(195,361)(196,360)(197,359)(198,358)(199,357)
(200,356);
s2 := Sym(400)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 25)( 12, 24)
( 13, 23)( 14, 22)( 15, 21)( 16, 20)( 17, 19)( 26, 31)( 27, 35)( 28, 34)
( 29, 33)( 30, 32)( 36, 50)( 37, 49)( 38, 48)( 39, 47)( 40, 46)( 41, 45)
( 42, 44)( 51, 81)( 52, 85)( 53, 84)( 54, 83)( 55, 82)( 56, 76)( 57, 80)
( 58, 79)( 59, 78)( 60, 77)( 61,100)( 62, 99)( 63, 98)( 64, 97)( 65, 96)
( 66, 95)( 67, 94)( 68, 93)( 69, 92)( 70, 91)( 71, 90)( 72, 89)( 73, 88)
( 74, 87)( 75, 86)(101,106)(102,110)(103,109)(104,108)(105,107)(111,125)
(112,124)(113,123)(114,122)(115,121)(116,120)(117,119)(126,131)(127,135)
(128,134)(129,133)(130,132)(136,150)(137,149)(138,148)(139,147)(140,146)
(141,145)(142,144)(151,181)(152,185)(153,184)(154,183)(155,182)(156,176)
(157,180)(158,179)(159,178)(160,177)(161,200)(162,199)(163,198)(164,197)
(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)
(173,188)(174,187)(175,186)(201,256)(202,260)(203,259)(204,258)(205,257)
(206,251)(207,255)(208,254)(209,253)(210,252)(211,275)(212,274)(213,273)
(214,272)(215,271)(216,270)(217,269)(218,268)(219,267)(220,266)(221,265)
(222,264)(223,263)(224,262)(225,261)(226,281)(227,285)(228,284)(229,283)
(230,282)(231,276)(232,280)(233,279)(234,278)(235,277)(236,300)(237,299)
(238,298)(239,297)(240,296)(241,295)(242,294)(243,293)(244,292)(245,291)
(246,290)(247,289)(248,288)(249,287)(250,286)(301,356)(302,360)(303,359)
(304,358)(305,357)(306,351)(307,355)(308,354)(309,353)(310,352)(311,375)
(312,374)(313,373)(314,372)(315,371)(316,370)(317,369)(318,368)(319,367)
(320,366)(321,365)(322,364)(323,363)(324,362)(325,361)(326,381)(327,385)
(328,384)(329,383)(330,382)(331,376)(332,380)(333,379)(334,378)(335,377)
(336,400)(337,399)(338,398)(339,397)(340,396)(341,395)(342,394)(343,393)
(344,392)(345,391)(346,390)(347,389)(348,388)(349,387)(350,386);
poly := sub<Sym(400)|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*s0*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*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*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*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