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# Polytope of Type {80,10}

Atlas Canonical Name : {80,10}*1600a
Also Known As : {80,10|2}. if this polytope has another name.
Group : SmallGroup(1600,2764)
Rank : 3
Schlafli Type : {80,10}
Number of vertices, edges, etc : 80, 400, 10
Order of s0s1s2 : 80
Order of s0s1s2s1 : 2
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}*800a
4-fold quotients : {20,10}*400a
5-fold quotients : {80,2}*320, {16,10}*320
8-fold quotients : {10,10}*200a
10-fold quotients : {40,2}*160, {8,10}*160
20-fold quotients : {20,2}*80, {4,10}*80
25-fold quotients : {16,2}*64
40-fold quotients : {2,10}*40, {10,2}*40
50-fold quotients : {8,2}*32
80-fold quotients : {2,5}*20, {5,2}*20
100-fold quotients : {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)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 51, 76)( 52, 80)( 53, 79)( 54, 78)
( 55, 77)( 56, 81)( 57, 85)( 58, 84)( 59, 83)( 60, 82)( 61, 86)( 62, 90)
( 63, 89)( 64, 88)( 65, 87)( 66, 91)( 67, 95)( 68, 94)( 69, 93)( 70, 92)
( 71, 96)( 72,100)( 73, 99)( 74, 98)( 75, 97)(101,151)(102,155)(103,154)
(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,161)
(112,165)(113,164)(114,163)(115,162)(116,166)(117,170)(118,169)(119,168)
(120,167)(121,171)(122,175)(123,174)(124,173)(125,172)(126,176)(127,180)
(128,179)(129,178)(130,177)(131,181)(132,185)(133,184)(134,183)(135,182)
(136,186)(137,190)(138,189)(139,188)(140,187)(141,191)(142,195)(143,194)
(144,193)(145,192)(146,196)(147,200)(148,199)(149,198)(150,197)(201,301)
(202,305)(203,304)(204,303)(205,302)(206,306)(207,310)(208,309)(209,308)
(210,307)(211,311)(212,315)(213,314)(214,313)(215,312)(216,316)(217,320)
(218,319)(219,318)(220,317)(221,321)(222,325)(223,324)(224,323)(225,322)
(226,326)(227,330)(228,329)(229,328)(230,327)(231,331)(232,335)(233,334)
(234,333)(235,332)(236,336)(237,340)(238,339)(239,338)(240,337)(241,341)
(242,345)(243,344)(244,343)(245,342)(246,346)(247,350)(248,349)(249,348)
(250,347)(251,376)(252,380)(253,379)(254,378)(255,377)(256,381)(257,385)
(258,384)(259,383)(260,382)(261,386)(262,390)(263,389)(264,388)(265,387)
(266,391)(267,395)(268,394)(269,393)(270,392)(271,396)(272,400)(273,399)
(274,398)(275,397)(276,351)(277,355)(278,354)(279,353)(280,352)(281,356)
(282,360)(283,359)(284,358)(285,357)(286,361)(287,365)(288,364)(289,363)
(290,362)(291,366)(292,370)(293,369)(294,368)(295,367)(296,371)(297,375)
(298,374)(299,373)(300,372);;
s1 := (  1,202)(  2,201)(  3,205)(  4,204)(  5,203)(  6,222)(  7,221)(  8,225)
(  9,224)( 10,223)( 11,217)( 12,216)( 13,220)( 14,219)( 15,218)( 16,212)
( 17,211)( 18,215)( 19,214)( 20,213)( 21,207)( 22,206)( 23,210)( 24,209)
( 25,208)( 26,227)( 27,226)( 28,230)( 29,229)( 30,228)( 31,247)( 32,246)
( 33,250)( 34,249)( 35,248)( 36,242)( 37,241)( 38,245)( 39,244)( 40,243)
( 41,237)( 42,236)( 43,240)( 44,239)( 45,238)( 46,232)( 47,231)( 48,235)
( 49,234)( 50,233)( 51,277)( 52,276)( 53,280)( 54,279)( 55,278)( 56,297)
( 57,296)( 58,300)( 59,299)( 60,298)( 61,292)( 62,291)( 63,295)( 64,294)
( 65,293)( 66,287)( 67,286)( 68,290)( 69,289)( 70,288)( 71,282)( 72,281)
( 73,285)( 74,284)( 75,283)( 76,252)( 77,251)( 78,255)( 79,254)( 80,253)
( 81,272)( 82,271)( 83,275)( 84,274)( 85,273)( 86,267)( 87,266)( 88,270)
( 89,269)( 90,268)( 91,262)( 92,261)( 93,265)( 94,264)( 95,263)( 96,257)
( 97,256)( 98,260)( 99,259)(100,258)(101,352)(102,351)(103,355)(104,354)
(105,353)(106,372)(107,371)(108,375)(109,374)(110,373)(111,367)(112,366)
(113,370)(114,369)(115,368)(116,362)(117,361)(118,365)(119,364)(120,363)
(121,357)(122,356)(123,360)(124,359)(125,358)(126,377)(127,376)(128,380)
(129,379)(130,378)(131,397)(132,396)(133,400)(134,399)(135,398)(136,392)
(137,391)(138,395)(139,394)(140,393)(141,387)(142,386)(143,390)(144,389)
(145,388)(146,382)(147,381)(148,385)(149,384)(150,383)(151,302)(152,301)
(153,305)(154,304)(155,303)(156,322)(157,321)(158,325)(159,324)(160,323)
(161,317)(162,316)(163,320)(164,319)(165,318)(166,312)(167,311)(168,315)
(169,314)(170,313)(171,307)(172,306)(173,310)(174,309)(175,308)(176,327)
(177,326)(178,330)(179,329)(180,328)(181,347)(182,346)(183,350)(184,349)
(185,348)(186,342)(187,341)(188,345)(189,344)(190,343)(191,337)(192,336)
(193,340)(194,339)(195,338)(196,332)(197,331)(198,335)(199,334)(200,333);;
s2 := (  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 11, 21)( 12, 22)( 13, 23)
( 14, 24)( 15, 25)( 26, 31)( 27, 32)( 28, 33)( 29, 34)( 30, 35)( 36, 46)
( 37, 47)( 38, 48)( 39, 49)( 40, 50)( 51, 56)( 52, 57)( 53, 58)( 54, 59)
( 55, 60)( 61, 71)( 62, 72)( 63, 73)( 64, 74)( 65, 75)( 76, 81)( 77, 82)
( 78, 83)( 79, 84)( 80, 85)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)
(101,106)(102,107)(103,108)(104,109)(105,110)(111,121)(112,122)(113,123)
(114,124)(115,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,146)
(137,147)(138,148)(139,149)(140,150)(151,156)(152,157)(153,158)(154,159)
(155,160)(161,171)(162,172)(163,173)(164,174)(165,175)(176,181)(177,182)
(178,183)(179,184)(180,185)(186,196)(187,197)(188,198)(189,199)(190,200)
(201,206)(202,207)(203,208)(204,209)(205,210)(211,221)(212,222)(213,223)
(214,224)(215,225)(226,231)(227,232)(228,233)(229,234)(230,235)(236,246)
(237,247)(238,248)(239,249)(240,250)(251,256)(252,257)(253,258)(254,259)
(255,260)(261,271)(262,272)(263,273)(264,274)(265,275)(276,281)(277,282)
(278,283)(279,284)(280,285)(286,296)(287,297)(288,298)(289,299)(290,300)
(301,306)(302,307)(303,308)(304,309)(305,310)(311,321)(312,322)(313,323)
(314,324)(315,325)(326,331)(327,332)(328,333)(329,334)(330,335)(336,346)
(337,347)(338,348)(339,349)(340,350)(351,356)(352,357)(353,358)(354,359)
(355,360)(361,371)(362,372)(363,373)(364,374)(365,375)(376,381)(377,382)
(378,383)(379,384)(380,385)(386,396)(387,397)(388,398)(389,399)(390,400);;
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*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,
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*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)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 51, 76)( 52, 80)( 53, 79)
( 54, 78)( 55, 77)( 56, 81)( 57, 85)( 58, 84)( 59, 83)( 60, 82)( 61, 86)
( 62, 90)( 63, 89)( 64, 88)( 65, 87)( 66, 91)( 67, 95)( 68, 94)( 69, 93)
( 70, 92)( 71, 96)( 72,100)( 73, 99)( 74, 98)( 75, 97)(101,151)(102,155)
(103,154)(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)
(111,161)(112,165)(113,164)(114,163)(115,162)(116,166)(117,170)(118,169)
(119,168)(120,167)(121,171)(122,175)(123,174)(124,173)(125,172)(126,176)
(127,180)(128,179)(129,178)(130,177)(131,181)(132,185)(133,184)(134,183)
(135,182)(136,186)(137,190)(138,189)(139,188)(140,187)(141,191)(142,195)
(143,194)(144,193)(145,192)(146,196)(147,200)(148,199)(149,198)(150,197)
(201,301)(202,305)(203,304)(204,303)(205,302)(206,306)(207,310)(208,309)
(209,308)(210,307)(211,311)(212,315)(213,314)(214,313)(215,312)(216,316)
(217,320)(218,319)(219,318)(220,317)(221,321)(222,325)(223,324)(224,323)
(225,322)(226,326)(227,330)(228,329)(229,328)(230,327)(231,331)(232,335)
(233,334)(234,333)(235,332)(236,336)(237,340)(238,339)(239,338)(240,337)
(241,341)(242,345)(243,344)(244,343)(245,342)(246,346)(247,350)(248,349)
(249,348)(250,347)(251,376)(252,380)(253,379)(254,378)(255,377)(256,381)
(257,385)(258,384)(259,383)(260,382)(261,386)(262,390)(263,389)(264,388)
(265,387)(266,391)(267,395)(268,394)(269,393)(270,392)(271,396)(272,400)
(273,399)(274,398)(275,397)(276,351)(277,355)(278,354)(279,353)(280,352)
(281,356)(282,360)(283,359)(284,358)(285,357)(286,361)(287,365)(288,364)
(289,363)(290,362)(291,366)(292,370)(293,369)(294,368)(295,367)(296,371)
(297,375)(298,374)(299,373)(300,372);
s1 := Sym(400)!(  1,202)(  2,201)(  3,205)(  4,204)(  5,203)(  6,222)(  7,221)
(  8,225)(  9,224)( 10,223)( 11,217)( 12,216)( 13,220)( 14,219)( 15,218)
( 16,212)( 17,211)( 18,215)( 19,214)( 20,213)( 21,207)( 22,206)( 23,210)
( 24,209)( 25,208)( 26,227)( 27,226)( 28,230)( 29,229)( 30,228)( 31,247)
( 32,246)( 33,250)( 34,249)( 35,248)( 36,242)( 37,241)( 38,245)( 39,244)
( 40,243)( 41,237)( 42,236)( 43,240)( 44,239)( 45,238)( 46,232)( 47,231)
( 48,235)( 49,234)( 50,233)( 51,277)( 52,276)( 53,280)( 54,279)( 55,278)
( 56,297)( 57,296)( 58,300)( 59,299)( 60,298)( 61,292)( 62,291)( 63,295)
( 64,294)( 65,293)( 66,287)( 67,286)( 68,290)( 69,289)( 70,288)( 71,282)
( 72,281)( 73,285)( 74,284)( 75,283)( 76,252)( 77,251)( 78,255)( 79,254)
( 80,253)( 81,272)( 82,271)( 83,275)( 84,274)( 85,273)( 86,267)( 87,266)
( 88,270)( 89,269)( 90,268)( 91,262)( 92,261)( 93,265)( 94,264)( 95,263)
( 96,257)( 97,256)( 98,260)( 99,259)(100,258)(101,352)(102,351)(103,355)
(104,354)(105,353)(106,372)(107,371)(108,375)(109,374)(110,373)(111,367)
(112,366)(113,370)(114,369)(115,368)(116,362)(117,361)(118,365)(119,364)
(120,363)(121,357)(122,356)(123,360)(124,359)(125,358)(126,377)(127,376)
(128,380)(129,379)(130,378)(131,397)(132,396)(133,400)(134,399)(135,398)
(136,392)(137,391)(138,395)(139,394)(140,393)(141,387)(142,386)(143,390)
(144,389)(145,388)(146,382)(147,381)(148,385)(149,384)(150,383)(151,302)
(152,301)(153,305)(154,304)(155,303)(156,322)(157,321)(158,325)(159,324)
(160,323)(161,317)(162,316)(163,320)(164,319)(165,318)(166,312)(167,311)
(168,315)(169,314)(170,313)(171,307)(172,306)(173,310)(174,309)(175,308)
(176,327)(177,326)(178,330)(179,329)(180,328)(181,347)(182,346)(183,350)
(184,349)(185,348)(186,342)(187,341)(188,345)(189,344)(190,343)(191,337)
(192,336)(193,340)(194,339)(195,338)(196,332)(197,331)(198,335)(199,334)
(200,333);
s2 := Sym(400)!(  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 11, 21)( 12, 22)
( 13, 23)( 14, 24)( 15, 25)( 26, 31)( 27, 32)( 28, 33)( 29, 34)( 30, 35)
( 36, 46)( 37, 47)( 38, 48)( 39, 49)( 40, 50)( 51, 56)( 52, 57)( 53, 58)
( 54, 59)( 55, 60)( 61, 71)( 62, 72)( 63, 73)( 64, 74)( 65, 75)( 76, 81)
( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 86, 96)( 87, 97)( 88, 98)( 89, 99)
( 90,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,121)(112,122)
(113,123)(114,124)(115,125)(126,131)(127,132)(128,133)(129,134)(130,135)
(136,146)(137,147)(138,148)(139,149)(140,150)(151,156)(152,157)(153,158)
(154,159)(155,160)(161,171)(162,172)(163,173)(164,174)(165,175)(176,181)
(177,182)(178,183)(179,184)(180,185)(186,196)(187,197)(188,198)(189,199)
(190,200)(201,206)(202,207)(203,208)(204,209)(205,210)(211,221)(212,222)
(213,223)(214,224)(215,225)(226,231)(227,232)(228,233)(229,234)(230,235)
(236,246)(237,247)(238,248)(239,249)(240,250)(251,256)(252,257)(253,258)
(254,259)(255,260)(261,271)(262,272)(263,273)(264,274)(265,275)(276,281)
(277,282)(278,283)(279,284)(280,285)(286,296)(287,297)(288,298)(289,299)
(290,300)(301,306)(302,307)(303,308)(304,309)(305,310)(311,321)(312,322)
(313,323)(314,324)(315,325)(326,331)(327,332)(328,333)(329,334)(330,335)
(336,346)(337,347)(338,348)(339,349)(340,350)(351,356)(352,357)(353,358)
(354,359)(355,360)(361,371)(362,372)(363,373)(364,374)(365,375)(376,381)
(377,382)(378,383)(379,384)(380,385)(386,396)(387,397)(388,398)(389,399)
(390,400);
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*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,
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*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