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# Polytope of Type {20,20}

Atlas Canonical Name : {20,20}*1600a
if this polytope has a name.
Group : SmallGroup(1600,2943)
Rank : 3
Schlafli Type : {20,20}
Number of vertices, edges, etc : 40, 400, 40
Order of s0s1s2 : 20
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
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 : {20,20}*800a
4-fold quotients : {10,20}*400a, {20,10}*400a
5-fold quotients : {4,20}*320, {20,4}*320
8-fold quotients : {10,10}*200a
10-fold quotients : {4,20}*160, {20,4}*160
20-fold quotients : {2,20}*80, {20,2}*80, {4,10}*80, {10,4}*80
25-fold quotients : {4,4}*64
40-fold quotients : {2,10}*40, {10,2}*40
50-fold quotients : {4,4}*32
80-fold quotients : {2,5}*20, {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 := (  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)
(101,126)(102,127)(103,128)(104,129)(105,130)(106,146)(107,147)(108,148)
(109,149)(110,150)(111,141)(112,142)(113,143)(114,144)(115,145)(116,136)
(117,137)(118,138)(119,139)(120,140)(121,131)(122,132)(123,133)(124,134)
(125,135)(151,176)(152,177)(153,178)(154,179)(155,180)(156,196)(157,197)
(158,198)(159,199)(160,200)(161,191)(162,192)(163,193)(164,194)(165,195)
(166,186)(167,187)(168,188)(169,189)(170,190)(171,181)(172,182)(173,183)
(174,184)(175,185)(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)(301,376)(302,377)(303,378)(304,379)
(305,380)(306,396)(307,397)(308,398)(309,399)(310,400)(311,391)(312,392)
(313,393)(314,394)(315,395)(316,386)(317,387)(318,388)(319,389)(320,390)
(321,381)(322,382)(323,383)(324,384)(325,385)(326,351)(327,352)(328,353)
(329,354)(330,355)(331,371)(332,372)(333,373)(334,374)(335,375)(336,366)
(337,367)(338,368)(339,369)(340,370)(341,361)(342,362)(343,363)(344,364)
(345,365)(346,356)(347,357)(348,358)(349,359)(350,360);;
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)(101,306)(102,310)(103,309)(104,308)
(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,321)(112,325)
(113,324)(114,323)(115,322)(116,316)(117,320)(118,319)(119,318)(120,317)
(121,311)(122,315)(123,314)(124,313)(125,312)(126,331)(127,335)(128,334)
(129,333)(130,332)(131,326)(132,330)(133,329)(134,328)(135,327)(136,346)
(137,350)(138,349)(139,348)(140,347)(141,341)(142,345)(143,344)(144,343)
(145,342)(146,336)(147,340)(148,339)(149,338)(150,337)(151,356)(152,360)
(153,359)(154,358)(155,357)(156,351)(157,355)(158,354)(159,353)(160,352)
(161,371)(162,375)(163,374)(164,373)(165,372)(166,366)(167,370)(168,369)
(169,368)(170,367)(171,361)(172,365)(173,364)(174,363)(175,362)(176,381)
(177,385)(178,384)(179,383)(180,382)(181,376)(182,380)(183,379)(184,378)
(185,377)(186,396)(187,400)(188,399)(189,398)(190,397)(191,391)(192,395)
(193,394)(194,393)(195,392)(196,386)(197,390)(198,389)(199,388)(200,387);;
s2 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 77)( 52, 76)( 53, 80)( 54, 79)
( 55, 78)( 56, 82)( 57, 81)( 58, 85)( 59, 84)( 60, 83)( 61, 87)( 62, 86)
( 63, 90)( 64, 89)( 65, 88)( 66, 92)( 67, 91)( 68, 95)( 69, 94)( 70, 93)
( 71, 97)( 72, 96)( 73,100)( 74, 99)( 75, 98)(101,102)(103,105)(106,107)
(108,110)(111,112)(113,115)(116,117)(118,120)(121,122)(123,125)(126,127)
(128,130)(131,132)(133,135)(136,137)(138,140)(141,142)(143,145)(146,147)
(148,150)(151,177)(152,176)(153,180)(154,179)(155,178)(156,182)(157,181)
(158,185)(159,184)(160,183)(161,187)(162,186)(163,190)(164,189)(165,188)
(166,192)(167,191)(168,195)(169,194)(170,193)(171,197)(172,196)(173,200)
(174,199)(175,198)(201,302)(202,301)(203,305)(204,304)(205,303)(206,307)
(207,306)(208,310)(209,309)(210,308)(211,312)(212,311)(213,315)(214,314)
(215,313)(216,317)(217,316)(218,320)(219,319)(220,318)(221,322)(222,321)
(223,325)(224,324)(225,323)(226,327)(227,326)(228,330)(229,329)(230,328)
(231,332)(232,331)(233,335)(234,334)(235,333)(236,337)(237,336)(238,340)
(239,339)(240,338)(241,342)(242,341)(243,345)(244,344)(245,343)(246,347)
(247,346)(248,350)(249,349)(250,348)(251,377)(252,376)(253,380)(254,379)
(255,378)(256,382)(257,381)(258,385)(259,384)(260,383)(261,387)(262,386)
(263,390)(264,389)(265,388)(266,392)(267,391)(268,395)(269,394)(270,393)
(271,397)(272,396)(273,400)(274,399)(275,398)(276,352)(277,351)(278,355)
(279,354)(280,353)(281,357)(282,356)(283,360)(284,359)(285,358)(286,362)
(287,361)(288,365)(289,364)(290,363)(291,367)(292,366)(293,370)(294,369)
(295,368)(296,372)(297,371)(298,375)(299,374)(300,373);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)(101,126)(102,127)(103,128)(104,129)(105,130)(106,146)(107,147)
(108,148)(109,149)(110,150)(111,141)(112,142)(113,143)(114,144)(115,145)
(116,136)(117,137)(118,138)(119,139)(120,140)(121,131)(122,132)(123,133)
(124,134)(125,135)(151,176)(152,177)(153,178)(154,179)(155,180)(156,196)
(157,197)(158,198)(159,199)(160,200)(161,191)(162,192)(163,193)(164,194)
(165,195)(166,186)(167,187)(168,188)(169,189)(170,190)(171,181)(172,182)
(173,183)(174,184)(175,185)(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)(301,376)(302,377)(303,378)
(304,379)(305,380)(306,396)(307,397)(308,398)(309,399)(310,400)(311,391)
(312,392)(313,393)(314,394)(315,395)(316,386)(317,387)(318,388)(319,389)
(320,390)(321,381)(322,382)(323,383)(324,384)(325,385)(326,351)(327,352)
(328,353)(329,354)(330,355)(331,371)(332,372)(333,373)(334,374)(335,375)
(336,366)(337,367)(338,368)(339,369)(340,370)(341,361)(342,362)(343,363)
(344,364)(345,365)(346,356)(347,357)(348,358)(349,359)(350,360);
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)(101,306)(102,310)(103,309)
(104,308)(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,321)
(112,325)(113,324)(114,323)(115,322)(116,316)(117,320)(118,319)(119,318)
(120,317)(121,311)(122,315)(123,314)(124,313)(125,312)(126,331)(127,335)
(128,334)(129,333)(130,332)(131,326)(132,330)(133,329)(134,328)(135,327)
(136,346)(137,350)(138,349)(139,348)(140,347)(141,341)(142,345)(143,344)
(144,343)(145,342)(146,336)(147,340)(148,339)(149,338)(150,337)(151,356)
(152,360)(153,359)(154,358)(155,357)(156,351)(157,355)(158,354)(159,353)
(160,352)(161,371)(162,375)(163,374)(164,373)(165,372)(166,366)(167,370)
(168,369)(169,368)(170,367)(171,361)(172,365)(173,364)(174,363)(175,362)
(176,381)(177,385)(178,384)(179,383)(180,382)(181,376)(182,380)(183,379)
(184,378)(185,377)(186,396)(187,400)(188,399)(189,398)(190,397)(191,391)
(192,395)(193,394)(194,393)(195,392)(196,386)(197,390)(198,389)(199,388)
(200,387);
s2 := Sym(400)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 77)( 52, 76)( 53, 80)
( 54, 79)( 55, 78)( 56, 82)( 57, 81)( 58, 85)( 59, 84)( 60, 83)( 61, 87)
( 62, 86)( 63, 90)( 64, 89)( 65, 88)( 66, 92)( 67, 91)( 68, 95)( 69, 94)
( 70, 93)( 71, 97)( 72, 96)( 73,100)( 74, 99)( 75, 98)(101,102)(103,105)
(106,107)(108,110)(111,112)(113,115)(116,117)(118,120)(121,122)(123,125)
(126,127)(128,130)(131,132)(133,135)(136,137)(138,140)(141,142)(143,145)
(146,147)(148,150)(151,177)(152,176)(153,180)(154,179)(155,178)(156,182)
(157,181)(158,185)(159,184)(160,183)(161,187)(162,186)(163,190)(164,189)
(165,188)(166,192)(167,191)(168,195)(169,194)(170,193)(171,197)(172,196)
(173,200)(174,199)(175,198)(201,302)(202,301)(203,305)(204,304)(205,303)
(206,307)(207,306)(208,310)(209,309)(210,308)(211,312)(212,311)(213,315)
(214,314)(215,313)(216,317)(217,316)(218,320)(219,319)(220,318)(221,322)
(222,321)(223,325)(224,324)(225,323)(226,327)(227,326)(228,330)(229,329)
(230,328)(231,332)(232,331)(233,335)(234,334)(235,333)(236,337)(237,336)
(238,340)(239,339)(240,338)(241,342)(242,341)(243,345)(244,344)(245,343)
(246,347)(247,346)(248,350)(249,349)(250,348)(251,377)(252,376)(253,380)
(254,379)(255,378)(256,382)(257,381)(258,385)(259,384)(260,383)(261,387)
(262,386)(263,390)(264,389)(265,388)(266,392)(267,391)(268,395)(269,394)
(270,393)(271,397)(272,396)(273,400)(274,399)(275,398)(276,352)(277,351)
(278,355)(279,354)(280,353)(281,357)(282,356)(283,360)(284,359)(285,358)
(286,362)(287,361)(288,365)(289,364)(290,363)(291,367)(292,366)(293,370)
(294,369)(295,368)(296,372)(297,371)(298,375)(299,374)(300,373);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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