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# Polytope of Type {4,80}

Atlas Canonical Name : {4,80}*640a
Also Known As : {4,80|2}. if this polytope has another name.
Group : SmallGroup(640,1983)
Rank : 3
Schlafli Type : {4,80}
Number of vertices, edges, etc : 4, 160, 80
Order of s0s1s2 : 80
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,80,2} of size 1280
Vertex Figure Of :
{2,4,80} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,40}*320a, {2,80}*320
4-fold quotients : {4,20}*160, {2,40}*160
5-fold quotients : {4,16}*128a
8-fold quotients : {2,20}*80, {4,10}*80
10-fold quotients : {4,8}*64a, {2,16}*64
16-fold quotients : {2,10}*40
20-fold quotients : {4,4}*32, {2,8}*32
32-fold quotients : {2,5}*20
40-fold quotients : {2,4}*16, {4,2}*16
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,80}*1280a, {8,80}*1280c, {8,80}*1280d, {4,160}*1280a, {4,160}*1280b
3-fold covers : {4,240}*1920a, {12,80}*1920a
Permutation Representation (GAP) :
```s0 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)( 24,104)
( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)
( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)
(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)
(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)
(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)
(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)
(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288)
(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)(216,296)
(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)
(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)
(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 11, 16)( 12, 20)( 13, 19)( 14, 18)
( 15, 17)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 31, 36)( 32, 40)( 33, 39)
( 34, 38)( 35, 37)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 46, 56)
( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 71)( 62, 75)( 63, 74)( 64, 73)
( 65, 72)( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,101)( 82,105)
( 83,104)( 84,103)( 85,102)( 86,106)( 87,110)( 88,109)( 89,108)( 90,107)
( 91,116)( 92,120)( 93,119)( 94,118)( 95,117)( 96,111)( 97,115)( 98,114)
( 99,113)(100,112)(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)
(127,160)(128,159)(129,158)(130,157)(131,141)(132,145)(133,144)(134,143)
(135,142)(136,146)(137,150)(138,149)(139,148)(140,147)(161,201)(162,205)
(163,204)(164,203)(165,202)(166,206)(167,210)(168,209)(169,208)(170,207)
(171,216)(172,220)(173,219)(174,218)(175,217)(176,211)(177,215)(178,214)
(179,213)(180,212)(181,221)(182,225)(183,224)(184,223)(185,222)(186,226)
(187,230)(188,229)(189,228)(190,227)(191,236)(192,240)(193,239)(194,238)
(195,237)(196,231)(197,235)(198,234)(199,233)(200,232)(241,301)(242,305)
(243,304)(244,303)(245,302)(246,306)(247,310)(248,309)(249,308)(250,307)
(251,316)(252,320)(253,319)(254,318)(255,317)(256,311)(257,315)(258,314)
(259,313)(260,312)(261,281)(262,285)(263,284)(264,283)(265,282)(266,286)
(267,290)(268,289)(269,288)(270,287)(271,296)(272,300)(273,299)(274,298)
(275,297)(276,291)(277,295)(278,294)(279,293)(280,292);;
s2 := (  1,162)(  2,161)(  3,165)(  4,164)(  5,163)(  6,167)(  7,166)(  8,170)
(  9,169)( 10,168)( 11,177)( 12,176)( 13,180)( 14,179)( 15,178)( 16,172)
( 17,171)( 18,175)( 19,174)( 20,173)( 21,182)( 22,181)( 23,185)( 24,184)
( 25,183)( 26,187)( 27,186)( 28,190)( 29,189)( 30,188)( 31,197)( 32,196)
( 33,200)( 34,199)( 35,198)( 36,192)( 37,191)( 38,195)( 39,194)( 40,193)
( 41,212)( 42,211)( 43,215)( 44,214)( 45,213)( 46,217)( 47,216)( 48,220)
( 49,219)( 50,218)( 51,202)( 52,201)( 53,205)( 54,204)( 55,203)( 56,207)
( 57,206)( 58,210)( 59,209)( 60,208)( 61,232)( 62,231)( 63,235)( 64,234)
( 65,233)( 66,237)( 67,236)( 68,240)( 69,239)( 70,238)( 71,222)( 72,221)
( 73,225)( 74,224)( 75,223)( 76,227)( 77,226)( 78,230)( 79,229)( 80,228)
( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,247)( 87,246)( 88,250)
( 89,249)( 90,248)( 91,257)( 92,256)( 93,260)( 94,259)( 95,258)( 96,252)
( 97,251)( 98,255)( 99,254)(100,253)(101,262)(102,261)(103,265)(104,264)
(105,263)(106,267)(107,266)(108,270)(109,269)(110,268)(111,277)(112,276)
(113,280)(114,279)(115,278)(116,272)(117,271)(118,275)(119,274)(120,273)
(121,292)(122,291)(123,295)(124,294)(125,293)(126,297)(127,296)(128,300)
(129,299)(130,298)(131,282)(132,281)(133,285)(134,284)(135,283)(136,287)
(137,286)(138,290)(139,289)(140,288)(141,312)(142,311)(143,315)(144,314)
(145,313)(146,317)(147,316)(148,320)(149,319)(150,318)(151,302)(152,301)
(153,305)(154,304)(155,303)(156,307)(157,306)(158,310)(159,309)(160,308);;
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(320)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)
( 24,104)( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)
( 48,128)( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)
( 56,136)( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)
( 72,152)( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)
( 80,160)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)
(168,248)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)
(176,256)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)
(184,264)(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)
(192,272)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)
(200,280)(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)
(208,288)(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)
(216,296)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)
(224,304)(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)
(232,312)(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)
(240,320);
s1 := Sym(320)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 11, 16)( 12, 20)( 13, 19)
( 14, 18)( 15, 17)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)
( 46, 56)( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 71)( 62, 75)( 63, 74)
( 64, 73)( 65, 72)( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,101)
( 82,105)( 83,104)( 84,103)( 85,102)( 86,106)( 87,110)( 88,109)( 89,108)
( 90,107)( 91,116)( 92,120)( 93,119)( 94,118)( 95,117)( 96,111)( 97,115)
( 98,114)( 99,113)(100,112)(121,151)(122,155)(123,154)(124,153)(125,152)
(126,156)(127,160)(128,159)(129,158)(130,157)(131,141)(132,145)(133,144)
(134,143)(135,142)(136,146)(137,150)(138,149)(139,148)(140,147)(161,201)
(162,205)(163,204)(164,203)(165,202)(166,206)(167,210)(168,209)(169,208)
(170,207)(171,216)(172,220)(173,219)(174,218)(175,217)(176,211)(177,215)
(178,214)(179,213)(180,212)(181,221)(182,225)(183,224)(184,223)(185,222)
(186,226)(187,230)(188,229)(189,228)(190,227)(191,236)(192,240)(193,239)
(194,238)(195,237)(196,231)(197,235)(198,234)(199,233)(200,232)(241,301)
(242,305)(243,304)(244,303)(245,302)(246,306)(247,310)(248,309)(249,308)
(250,307)(251,316)(252,320)(253,319)(254,318)(255,317)(256,311)(257,315)
(258,314)(259,313)(260,312)(261,281)(262,285)(263,284)(264,283)(265,282)
(266,286)(267,290)(268,289)(269,288)(270,287)(271,296)(272,300)(273,299)
(274,298)(275,297)(276,291)(277,295)(278,294)(279,293)(280,292);
s2 := Sym(320)!(  1,162)(  2,161)(  3,165)(  4,164)(  5,163)(  6,167)(  7,166)
(  8,170)(  9,169)( 10,168)( 11,177)( 12,176)( 13,180)( 14,179)( 15,178)
( 16,172)( 17,171)( 18,175)( 19,174)( 20,173)( 21,182)( 22,181)( 23,185)
( 24,184)( 25,183)( 26,187)( 27,186)( 28,190)( 29,189)( 30,188)( 31,197)
( 32,196)( 33,200)( 34,199)( 35,198)( 36,192)( 37,191)( 38,195)( 39,194)
( 40,193)( 41,212)( 42,211)( 43,215)( 44,214)( 45,213)( 46,217)( 47,216)
( 48,220)( 49,219)( 50,218)( 51,202)( 52,201)( 53,205)( 54,204)( 55,203)
( 56,207)( 57,206)( 58,210)( 59,209)( 60,208)( 61,232)( 62,231)( 63,235)
( 64,234)( 65,233)( 66,237)( 67,236)( 68,240)( 69,239)( 70,238)( 71,222)
( 72,221)( 73,225)( 74,224)( 75,223)( 76,227)( 77,226)( 78,230)( 79,229)
( 80,228)( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,247)( 87,246)
( 88,250)( 89,249)( 90,248)( 91,257)( 92,256)( 93,260)( 94,259)( 95,258)
( 96,252)( 97,251)( 98,255)( 99,254)(100,253)(101,262)(102,261)(103,265)
(104,264)(105,263)(106,267)(107,266)(108,270)(109,269)(110,268)(111,277)
(112,276)(113,280)(114,279)(115,278)(116,272)(117,271)(118,275)(119,274)
(120,273)(121,292)(122,291)(123,295)(124,294)(125,293)(126,297)(127,296)
(128,300)(129,299)(130,298)(131,282)(132,281)(133,285)(134,284)(135,283)
(136,287)(137,286)(138,290)(139,289)(140,288)(141,312)(142,311)(143,315)
(144,314)(145,313)(146,317)(147,316)(148,320)(149,319)(150,318)(151,302)
(152,301)(153,305)(154,304)(155,303)(156,307)(157,306)(158,310)(159,309)
(160,308);
poly := sub<Sym(320)|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 >;

```
References : None.
to this polytope