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# Polytope of Type {40,20}

Atlas Canonical Name : {40,20}*1600d
if this polytope has a name.
Group : SmallGroup(1600,3473)
Rank : 3
Schlafli Type : {40,20}
Number of vertices, edges, etc : 40, 400, 20
Order of s0s1s2 : 40
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {40,10}*800b, {20,20}*800c
4-fold quotients : {20,10}*400b, {10,20}*400c
5-fold quotients : {40,4}*320a
8-fold quotients : {10,10}*200c
10-fold quotients : {20,4}*160, {40,2}*160
16-fold quotients : {5,10}*100
20-fold quotients : {20,2}*80, {10,4}*80
25-fold quotients : {8,4}*64a
40-fold quotients : {10,2}*40
50-fold quotients : {4,4}*32, {8,2}*32
80-fold quotients : {5,2}*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 := (  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)( 11, 16)
( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 27, 30)( 28, 29)( 31, 46)( 32, 50)
( 33, 49)( 34, 48)( 35, 47)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)
( 52, 55)( 53, 54)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 66)
( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 77, 80)( 78, 79)( 81, 96)( 82,100)
( 83, 99)( 84, 98)( 85, 97)( 86, 91)( 87, 95)( 88, 94)( 89, 93)( 90, 92)
(101,126)(102,130)(103,129)(104,128)(105,127)(106,146)(107,150)(108,149)
(109,148)(110,147)(111,141)(112,145)(113,144)(114,143)(115,142)(116,136)
(117,140)(118,139)(119,138)(120,137)(121,131)(122,135)(123,134)(124,133)
(125,132)(151,176)(152,180)(153,179)(154,178)(155,177)(156,196)(157,200)
(158,199)(159,198)(160,197)(161,191)(162,195)(163,194)(164,193)(165,192)
(166,186)(167,190)(168,189)(169,188)(170,187)(171,181)(172,185)(173,184)
(174,183)(175,182)(201,351)(202,355)(203,354)(204,353)(205,352)(206,371)
(207,375)(208,374)(209,373)(210,372)(211,366)(212,370)(213,369)(214,368)
(215,367)(216,361)(217,365)(218,364)(219,363)(220,362)(221,356)(222,360)
(223,359)(224,358)(225,357)(226,376)(227,380)(228,379)(229,378)(230,377)
(231,396)(232,400)(233,399)(234,398)(235,397)(236,391)(237,395)(238,394)
(239,393)(240,392)(241,386)(242,390)(243,389)(244,388)(245,387)(246,381)
(247,385)(248,384)(249,383)(250,382)(251,301)(252,305)(253,304)(254,303)
(255,302)(256,321)(257,325)(258,324)(259,323)(260,322)(261,316)(262,320)
(263,319)(264,318)(265,317)(266,311)(267,315)(268,314)(269,313)(270,312)
(271,306)(272,310)(273,309)(274,308)(275,307)(276,326)(277,330)(278,329)
(279,328)(280,327)(281,346)(282,350)(283,349)(284,348)(285,347)(286,341)
(287,345)(288,344)(289,343)(290,342)(291,336)(292,340)(293,339)(294,338)
(295,337)(296,331)(297,335)(298,334)(299,333)(300,332);;
s1 := (  1,207)(  2,206)(  3,210)(  4,209)(  5,208)(  6,202)(  7,201)(  8,205)
(  9,204)( 10,203)( 11,222)( 12,221)( 13,225)( 14,224)( 15,223)( 16,217)
( 17,216)( 18,220)( 19,219)( 20,218)( 21,212)( 22,211)( 23,215)( 24,214)
( 25,213)( 26,232)( 27,231)( 28,235)( 29,234)( 30,233)( 31,227)( 32,226)
( 33,230)( 34,229)( 35,228)( 36,247)( 37,246)( 38,250)( 39,249)( 40,248)
( 41,242)( 42,241)( 43,245)( 44,244)( 45,243)( 46,237)( 47,236)( 48,240)
( 49,239)( 50,238)( 51,257)( 52,256)( 53,260)( 54,259)( 55,258)( 56,252)
( 57,251)( 58,255)( 59,254)( 60,253)( 61,272)( 62,271)( 63,275)( 64,274)
( 65,273)( 66,267)( 67,266)( 68,270)( 69,269)( 70,268)( 71,262)( 72,261)
( 73,265)( 74,264)( 75,263)( 76,282)( 77,281)( 78,285)( 79,284)( 80,283)
( 81,277)( 82,276)( 83,280)( 84,279)( 85,278)( 86,297)( 87,296)( 88,300)
( 89,299)( 90,298)( 91,292)( 92,291)( 93,295)( 94,294)( 95,293)( 96,287)
( 97,286)( 98,290)( 99,289)(100,288)(101,332)(102,331)(103,335)(104,334)
(105,333)(106,327)(107,326)(108,330)(109,329)(110,328)(111,347)(112,346)
(113,350)(114,349)(115,348)(116,342)(117,341)(118,345)(119,344)(120,343)
(121,337)(122,336)(123,340)(124,339)(125,338)(126,307)(127,306)(128,310)
(129,309)(130,308)(131,302)(132,301)(133,305)(134,304)(135,303)(136,322)
(137,321)(138,325)(139,324)(140,323)(141,317)(142,316)(143,320)(144,319)
(145,318)(146,312)(147,311)(148,315)(149,314)(150,313)(151,382)(152,381)
(153,385)(154,384)(155,383)(156,377)(157,376)(158,380)(159,379)(160,378)
(161,397)(162,396)(163,400)(164,399)(165,398)(166,392)(167,391)(168,395)
(169,394)(170,393)(171,387)(172,386)(173,390)(174,389)(175,388)(176,357)
(177,356)(178,360)(179,359)(180,358)(181,352)(182,351)(183,355)(184,354)
(185,353)(186,372)(187,371)(188,375)(189,374)(190,373)(191,367)(192,366)
(193,370)(194,369)(195,368)(196,362)(197,361)(198,365)(199,364)(200,363);;
s2 := (  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);;
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*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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(400)!(  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)
( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 27, 30)( 28, 29)( 31, 46)
( 32, 50)( 33, 49)( 34, 48)( 35, 47)( 36, 41)( 37, 45)( 38, 44)( 39, 43)
( 40, 42)( 52, 55)( 53, 54)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)
( 61, 66)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 77, 80)( 78, 79)( 81, 96)
( 82,100)( 83, 99)( 84, 98)( 85, 97)( 86, 91)( 87, 95)( 88, 94)( 89, 93)
( 90, 92)(101,126)(102,130)(103,129)(104,128)(105,127)(106,146)(107,150)
(108,149)(109,148)(110,147)(111,141)(112,145)(113,144)(114,143)(115,142)
(116,136)(117,140)(118,139)(119,138)(120,137)(121,131)(122,135)(123,134)
(124,133)(125,132)(151,176)(152,180)(153,179)(154,178)(155,177)(156,196)
(157,200)(158,199)(159,198)(160,197)(161,191)(162,195)(163,194)(164,193)
(165,192)(166,186)(167,190)(168,189)(169,188)(170,187)(171,181)(172,185)
(173,184)(174,183)(175,182)(201,351)(202,355)(203,354)(204,353)(205,352)
(206,371)(207,375)(208,374)(209,373)(210,372)(211,366)(212,370)(213,369)
(214,368)(215,367)(216,361)(217,365)(218,364)(219,363)(220,362)(221,356)
(222,360)(223,359)(224,358)(225,357)(226,376)(227,380)(228,379)(229,378)
(230,377)(231,396)(232,400)(233,399)(234,398)(235,397)(236,391)(237,395)
(238,394)(239,393)(240,392)(241,386)(242,390)(243,389)(244,388)(245,387)
(246,381)(247,385)(248,384)(249,383)(250,382)(251,301)(252,305)(253,304)
(254,303)(255,302)(256,321)(257,325)(258,324)(259,323)(260,322)(261,316)
(262,320)(263,319)(264,318)(265,317)(266,311)(267,315)(268,314)(269,313)
(270,312)(271,306)(272,310)(273,309)(274,308)(275,307)(276,326)(277,330)
(278,329)(279,328)(280,327)(281,346)(282,350)(283,349)(284,348)(285,347)
(286,341)(287,345)(288,344)(289,343)(290,342)(291,336)(292,340)(293,339)
(294,338)(295,337)(296,331)(297,335)(298,334)(299,333)(300,332);
s1 := Sym(400)!(  1,207)(  2,206)(  3,210)(  4,209)(  5,208)(  6,202)(  7,201)
(  8,205)(  9,204)( 10,203)( 11,222)( 12,221)( 13,225)( 14,224)( 15,223)
( 16,217)( 17,216)( 18,220)( 19,219)( 20,218)( 21,212)( 22,211)( 23,215)
( 24,214)( 25,213)( 26,232)( 27,231)( 28,235)( 29,234)( 30,233)( 31,227)
( 32,226)( 33,230)( 34,229)( 35,228)( 36,247)( 37,246)( 38,250)( 39,249)
( 40,248)( 41,242)( 42,241)( 43,245)( 44,244)( 45,243)( 46,237)( 47,236)
( 48,240)( 49,239)( 50,238)( 51,257)( 52,256)( 53,260)( 54,259)( 55,258)
( 56,252)( 57,251)( 58,255)( 59,254)( 60,253)( 61,272)( 62,271)( 63,275)
( 64,274)( 65,273)( 66,267)( 67,266)( 68,270)( 69,269)( 70,268)( 71,262)
( 72,261)( 73,265)( 74,264)( 75,263)( 76,282)( 77,281)( 78,285)( 79,284)
( 80,283)( 81,277)( 82,276)( 83,280)( 84,279)( 85,278)( 86,297)( 87,296)
( 88,300)( 89,299)( 90,298)( 91,292)( 92,291)( 93,295)( 94,294)( 95,293)
( 96,287)( 97,286)( 98,290)( 99,289)(100,288)(101,332)(102,331)(103,335)
(104,334)(105,333)(106,327)(107,326)(108,330)(109,329)(110,328)(111,347)
(112,346)(113,350)(114,349)(115,348)(116,342)(117,341)(118,345)(119,344)
(120,343)(121,337)(122,336)(123,340)(124,339)(125,338)(126,307)(127,306)
(128,310)(129,309)(130,308)(131,302)(132,301)(133,305)(134,304)(135,303)
(136,322)(137,321)(138,325)(139,324)(140,323)(141,317)(142,316)(143,320)
(144,319)(145,318)(146,312)(147,311)(148,315)(149,314)(150,313)(151,382)
(152,381)(153,385)(154,384)(155,383)(156,377)(157,376)(158,380)(159,379)
(160,378)(161,397)(162,396)(163,400)(164,399)(165,398)(166,392)(167,391)
(168,395)(169,394)(170,393)(171,387)(172,386)(173,390)(174,389)(175,388)
(176,357)(177,356)(178,360)(179,359)(180,358)(181,352)(182,351)(183,355)
(184,354)(185,353)(186,372)(187,371)(188,375)(189,374)(190,373)(191,367)
(192,366)(193,370)(194,369)(195,368)(196,362)(197,361)(198,365)(199,364)
(200,363);
s2 := 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);
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*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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