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# Polytope of Type {20,40}

Atlas Canonical Name : {20,40}*1600d
if this polytope has a name.
Group : SmallGroup(1600,3473)
Rank : 3
Schlafli Type : {20,40}
Number of vertices, edges, etc : 20, 400, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 10
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 : {10,40}*800b, {20,20}*800b
4-fold quotients : {10,20}*400b, {20,10}*400c
5-fold quotients : {4,40}*320a
8-fold quotients : {10,10}*200b
10-fold quotients : {4,20}*160, {2,40}*160
16-fold quotients : {10,5}*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 := (  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)( 13, 18)
( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 41)
( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)( 59, 74)
( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)( 82, 97)
( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)( 90, 95)
(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)(113,118)
(114,119)(115,120)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)
(137,142)(138,143)(139,144)(140,145)(156,171)(157,172)(158,173)(159,174)
(160,175)(161,166)(162,167)(163,168)(164,169)(165,170)(181,196)(182,197)
(183,198)(184,199)(185,200)(186,191)(187,192)(188,193)(189,194)(190,195)
(201,251)(202,252)(203,253)(204,254)(205,255)(206,271)(207,272)(208,273)
(209,274)(210,275)(211,266)(212,267)(213,268)(214,269)(215,270)(216,261)
(217,262)(218,263)(219,264)(220,265)(221,256)(222,257)(223,258)(224,259)
(225,260)(226,276)(227,277)(228,278)(229,279)(230,280)(231,296)(232,297)
(233,298)(234,299)(235,300)(236,291)(237,292)(238,293)(239,294)(240,295)
(241,286)(242,287)(243,288)(244,289)(245,290)(246,281)(247,282)(248,283)
(249,284)(250,285)(301,351)(302,352)(303,353)(304,354)(305,355)(306,371)
(307,372)(308,373)(309,374)(310,375)(311,366)(312,367)(313,368)(314,369)
(315,370)(316,361)(317,362)(318,363)(319,364)(320,365)(321,356)(322,357)
(323,358)(324,359)(325,360)(326,376)(327,377)(328,378)(329,379)(330,380)
(331,396)(332,397)(333,398)(334,399)(335,400)(336,391)(337,392)(338,393)
(339,394)(340,395)(341,386)(342,387)(343,388)(344,389)(345,390)(346,381)
(347,382)(348,383)(349,384)(350,385);;
s1 := (  1,206)(  2,210)(  3,209)(  4,208)(  5,207)(  6,201)(  7,205)(  8,204)
(  9,203)( 10,202)( 11,221)( 12,225)( 13,224)( 14,223)( 15,222)( 16,216)
( 17,220)( 18,219)( 19,218)( 20,217)( 21,211)( 22,215)( 23,214)( 24,213)
( 25,212)( 26,231)( 27,235)( 28,234)( 29,233)( 30,232)( 31,226)( 32,230)
( 33,229)( 34,228)( 35,227)( 36,246)( 37,250)( 38,249)( 39,248)( 40,247)
( 41,241)( 42,245)( 43,244)( 44,243)( 45,242)( 46,236)( 47,240)( 48,239)
( 49,238)( 50,237)( 51,256)( 52,260)( 53,259)( 54,258)( 55,257)( 56,251)
( 57,255)( 58,254)( 59,253)( 60,252)( 61,271)( 62,275)( 63,274)( 64,273)
( 65,272)( 66,266)( 67,270)( 68,269)( 69,268)( 70,267)( 71,261)( 72,265)
( 73,264)( 74,263)( 75,262)( 76,281)( 77,285)( 78,284)( 79,283)( 80,282)
( 81,276)( 82,280)( 83,279)( 84,278)( 85,277)( 86,296)( 87,300)( 88,299)
( 89,298)( 90,297)( 91,291)( 92,295)( 93,294)( 94,293)( 95,292)( 96,286)
( 97,290)( 98,289)( 99,288)(100,287)(101,331)(102,335)(103,334)(104,333)
(105,332)(106,326)(107,330)(108,329)(109,328)(110,327)(111,346)(112,350)
(113,349)(114,348)(115,347)(116,341)(117,345)(118,344)(119,343)(120,342)
(121,336)(122,340)(123,339)(124,338)(125,337)(126,306)(127,310)(128,309)
(129,308)(130,307)(131,301)(132,305)(133,304)(134,303)(135,302)(136,321)
(137,325)(138,324)(139,323)(140,322)(141,316)(142,320)(143,319)(144,318)
(145,317)(146,311)(147,315)(148,314)(149,313)(150,312)(151,381)(152,385)
(153,384)(154,383)(155,382)(156,376)(157,380)(158,379)(159,378)(160,377)
(161,396)(162,400)(163,399)(164,398)(165,397)(166,391)(167,395)(168,394)
(169,393)(170,392)(171,386)(172,390)(173,389)(174,388)(175,387)(176,356)
(177,360)(178,359)(179,358)(180,357)(181,351)(182,355)(183,354)(184,353)
(185,352)(186,371)(187,375)(188,374)(189,373)(190,372)(191,366)(192,370)
(193,369)(194,368)(195,367)(196,361)(197,365)(198,364)(199,363)(200,362);;
s2 := (  1,  2)(  3,  5)(  6, 22)(  7, 21)(  8, 25)(  9, 24)( 10, 23)( 11, 17)
( 12, 16)( 13, 20)( 14, 19)( 15, 18)( 26, 27)( 28, 30)( 31, 47)( 32, 46)
( 33, 50)( 34, 49)( 35, 48)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)
( 51, 52)( 53, 55)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)( 61, 67)
( 62, 66)( 63, 70)( 64, 69)( 65, 68)( 76, 77)( 78, 80)( 81, 97)( 82, 96)
( 83,100)( 84, 99)( 85, 98)( 86, 92)( 87, 91)( 88, 95)( 89, 94)( 90, 93)
(101,127)(102,126)(103,130)(104,129)(105,128)(106,147)(107,146)(108,150)
(109,149)(110,148)(111,142)(112,141)(113,145)(114,144)(115,143)(116,137)
(117,136)(118,140)(119,139)(120,138)(121,132)(122,131)(123,135)(124,134)
(125,133)(151,177)(152,176)(153,180)(154,179)(155,178)(156,197)(157,196)
(158,200)(159,199)(160,198)(161,192)(162,191)(163,195)(164,194)(165,193)
(166,187)(167,186)(168,190)(169,189)(170,188)(171,182)(172,181)(173,185)
(174,184)(175,183)(201,352)(202,351)(203,355)(204,354)(205,353)(206,372)
(207,371)(208,375)(209,374)(210,373)(211,367)(212,366)(213,370)(214,369)
(215,368)(216,362)(217,361)(218,365)(219,364)(220,363)(221,357)(222,356)
(223,360)(224,359)(225,358)(226,377)(227,376)(228,380)(229,379)(230,378)
(231,397)(232,396)(233,400)(234,399)(235,398)(236,392)(237,391)(238,395)
(239,394)(240,393)(241,387)(242,386)(243,390)(244,389)(245,388)(246,382)
(247,381)(248,385)(249,384)(250,383)(251,302)(252,301)(253,305)(254,304)
(255,303)(256,322)(257,321)(258,325)(259,324)(260,323)(261,317)(262,316)
(263,320)(264,319)(265,318)(266,312)(267,311)(268,315)(269,314)(270,313)
(271,307)(272,306)(273,310)(274,309)(275,308)(276,327)(277,326)(278,330)
(279,329)(280,328)(281,347)(282,346)(283,350)(284,349)(285,348)(286,342)
(287,341)(288,345)(289,344)(290,343)(291,337)(292,336)(293,340)(294,339)
(295,338)(296,332)(297,331)(298,335)(299,334)(300,333);;
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, s2*s0*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)
( 13, 18)( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)
( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)
( 59, 74)( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)
( 82, 97)( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)
( 90, 95)(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)
(113,118)(114,119)(115,120)(131,146)(132,147)(133,148)(134,149)(135,150)
(136,141)(137,142)(138,143)(139,144)(140,145)(156,171)(157,172)(158,173)
(159,174)(160,175)(161,166)(162,167)(163,168)(164,169)(165,170)(181,196)
(182,197)(183,198)(184,199)(185,200)(186,191)(187,192)(188,193)(189,194)
(190,195)(201,251)(202,252)(203,253)(204,254)(205,255)(206,271)(207,272)
(208,273)(209,274)(210,275)(211,266)(212,267)(213,268)(214,269)(215,270)
(216,261)(217,262)(218,263)(219,264)(220,265)(221,256)(222,257)(223,258)
(224,259)(225,260)(226,276)(227,277)(228,278)(229,279)(230,280)(231,296)
(232,297)(233,298)(234,299)(235,300)(236,291)(237,292)(238,293)(239,294)
(240,295)(241,286)(242,287)(243,288)(244,289)(245,290)(246,281)(247,282)
(248,283)(249,284)(250,285)(301,351)(302,352)(303,353)(304,354)(305,355)
(306,371)(307,372)(308,373)(309,374)(310,375)(311,366)(312,367)(313,368)
(314,369)(315,370)(316,361)(317,362)(318,363)(319,364)(320,365)(321,356)
(322,357)(323,358)(324,359)(325,360)(326,376)(327,377)(328,378)(329,379)
(330,380)(331,396)(332,397)(333,398)(334,399)(335,400)(336,391)(337,392)
(338,393)(339,394)(340,395)(341,386)(342,387)(343,388)(344,389)(345,390)
(346,381)(347,382)(348,383)(349,384)(350,385);
s1 := Sym(400)!(  1,206)(  2,210)(  3,209)(  4,208)(  5,207)(  6,201)(  7,205)
(  8,204)(  9,203)( 10,202)( 11,221)( 12,225)( 13,224)( 14,223)( 15,222)
( 16,216)( 17,220)( 18,219)( 19,218)( 20,217)( 21,211)( 22,215)( 23,214)
( 24,213)( 25,212)( 26,231)( 27,235)( 28,234)( 29,233)( 30,232)( 31,226)
( 32,230)( 33,229)( 34,228)( 35,227)( 36,246)( 37,250)( 38,249)( 39,248)
( 40,247)( 41,241)( 42,245)( 43,244)( 44,243)( 45,242)( 46,236)( 47,240)
( 48,239)( 49,238)( 50,237)( 51,256)( 52,260)( 53,259)( 54,258)( 55,257)
( 56,251)( 57,255)( 58,254)( 59,253)( 60,252)( 61,271)( 62,275)( 63,274)
( 64,273)( 65,272)( 66,266)( 67,270)( 68,269)( 69,268)( 70,267)( 71,261)
( 72,265)( 73,264)( 74,263)( 75,262)( 76,281)( 77,285)( 78,284)( 79,283)
( 80,282)( 81,276)( 82,280)( 83,279)( 84,278)( 85,277)( 86,296)( 87,300)
( 88,299)( 89,298)( 90,297)( 91,291)( 92,295)( 93,294)( 94,293)( 95,292)
( 96,286)( 97,290)( 98,289)( 99,288)(100,287)(101,331)(102,335)(103,334)
(104,333)(105,332)(106,326)(107,330)(108,329)(109,328)(110,327)(111,346)
(112,350)(113,349)(114,348)(115,347)(116,341)(117,345)(118,344)(119,343)
(120,342)(121,336)(122,340)(123,339)(124,338)(125,337)(126,306)(127,310)
(128,309)(129,308)(130,307)(131,301)(132,305)(133,304)(134,303)(135,302)
(136,321)(137,325)(138,324)(139,323)(140,322)(141,316)(142,320)(143,319)
(144,318)(145,317)(146,311)(147,315)(148,314)(149,313)(150,312)(151,381)
(152,385)(153,384)(154,383)(155,382)(156,376)(157,380)(158,379)(159,378)
(160,377)(161,396)(162,400)(163,399)(164,398)(165,397)(166,391)(167,395)
(168,394)(169,393)(170,392)(171,386)(172,390)(173,389)(174,388)(175,387)
(176,356)(177,360)(178,359)(179,358)(180,357)(181,351)(182,355)(183,354)
(184,353)(185,352)(186,371)(187,375)(188,374)(189,373)(190,372)(191,366)
(192,370)(193,369)(194,368)(195,367)(196,361)(197,365)(198,364)(199,363)
(200,362);
s2 := Sym(400)!(  1,  2)(  3,  5)(  6, 22)(  7, 21)(  8, 25)(  9, 24)( 10, 23)
( 11, 17)( 12, 16)( 13, 20)( 14, 19)( 15, 18)( 26, 27)( 28, 30)( 31, 47)
( 32, 46)( 33, 50)( 34, 49)( 35, 48)( 36, 42)( 37, 41)( 38, 45)( 39, 44)
( 40, 43)( 51, 52)( 53, 55)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)
( 61, 67)( 62, 66)( 63, 70)( 64, 69)( 65, 68)( 76, 77)( 78, 80)( 81, 97)
( 82, 96)( 83,100)( 84, 99)( 85, 98)( 86, 92)( 87, 91)( 88, 95)( 89, 94)
( 90, 93)(101,127)(102,126)(103,130)(104,129)(105,128)(106,147)(107,146)
(108,150)(109,149)(110,148)(111,142)(112,141)(113,145)(114,144)(115,143)
(116,137)(117,136)(118,140)(119,139)(120,138)(121,132)(122,131)(123,135)
(124,134)(125,133)(151,177)(152,176)(153,180)(154,179)(155,178)(156,197)
(157,196)(158,200)(159,199)(160,198)(161,192)(162,191)(163,195)(164,194)
(165,193)(166,187)(167,186)(168,190)(169,189)(170,188)(171,182)(172,181)
(173,185)(174,184)(175,183)(201,352)(202,351)(203,355)(204,354)(205,353)
(206,372)(207,371)(208,375)(209,374)(210,373)(211,367)(212,366)(213,370)
(214,369)(215,368)(216,362)(217,361)(218,365)(219,364)(220,363)(221,357)
(222,356)(223,360)(224,359)(225,358)(226,377)(227,376)(228,380)(229,379)
(230,378)(231,397)(232,396)(233,400)(234,399)(235,398)(236,392)(237,391)
(238,395)(239,394)(240,393)(241,387)(242,386)(243,390)(244,389)(245,388)
(246,382)(247,381)(248,385)(249,384)(250,383)(251,302)(252,301)(253,305)
(254,304)(255,303)(256,322)(257,321)(258,325)(259,324)(260,323)(261,317)
(262,316)(263,320)(264,319)(265,318)(266,312)(267,311)(268,315)(269,314)
(270,313)(271,307)(272,306)(273,310)(274,309)(275,308)(276,327)(277,326)
(278,330)(279,329)(280,328)(281,347)(282,346)(283,350)(284,349)(285,348)
(286,342)(287,341)(288,345)(289,344)(290,343)(291,337)(292,336)(293,340)
(294,339)(295,338)(296,332)(297,331)(298,335)(299,334)(300,333);
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, s2*s0*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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