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# Polytope of Type {80,4}

Atlas Canonical Name : {80,4}*640b
if this polytope has a name.
Group : SmallGroup(640,1995)
Rank : 3
Schlafli Type : {80,4}
Number of vertices, edges, etc : 80, 160, 4
Order of s0s1s2 : 80
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{80,4,2} of size 1280
Vertex Figure Of :
{2,80,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {40,4}*320a
4-fold quotients : {20,4}*160, {40,2}*160
5-fold quotients : {16,4}*128b
8-fold quotients : {20,2}*80, {10,4}*80
10-fold quotients : {8,4}*64a
16-fold quotients : {10,2}*40
20-fold quotients : {4,4}*32, {8,2}*32
32-fold quotients : {5,2}*20
40-fold quotients : {2,4}*16, {4,2}*16
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {80,4}*1280a, {80,8}*1280e, {80,8}*1280f
3-fold covers : {240,4}*1920b, {80,12}*1920b
Permutation Representation (GAP) :
```s0 := (  1,161)(  2,165)(  3,164)(  4,163)(  5,162)(  6,166)(  7,170)(  8,169)
(  9,168)( 10,167)( 11,176)( 12,180)( 13,179)( 14,178)( 15,177)( 16,171)
( 17,175)( 18,174)( 19,173)( 20,172)( 21,186)( 22,190)( 23,189)( 24,188)
( 25,187)( 26,181)( 27,185)( 28,184)( 29,183)( 30,182)( 31,191)( 32,195)
( 33,194)( 34,193)( 35,192)( 36,196)( 37,200)( 38,199)( 39,198)( 40,197)
( 41,211)( 42,215)( 43,214)( 44,213)( 45,212)( 46,216)( 47,220)( 48,219)
( 49,218)( 50,217)( 51,201)( 52,205)( 53,204)( 54,203)( 55,202)( 56,206)
( 57,210)( 58,209)( 59,208)( 60,207)( 61,236)( 62,240)( 63,239)( 64,238)
( 65,237)( 66,231)( 67,235)( 68,234)( 69,233)( 70,232)( 71,226)( 72,230)
( 73,229)( 74,228)( 75,227)( 76,221)( 77,225)( 78,224)( 79,223)( 80,222)
( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,256)( 92,260)( 93,259)( 94,258)( 95,257)( 96,251)
( 97,255)( 98,254)( 99,253)(100,252)(101,266)(102,270)(103,269)(104,268)
(105,267)(106,261)(107,265)(108,264)(109,263)(110,262)(111,271)(112,275)
(113,274)(114,273)(115,272)(116,276)(117,280)(118,279)(119,278)(120,277)
(121,291)(122,295)(123,294)(124,293)(125,292)(126,296)(127,300)(128,299)
(129,298)(130,297)(131,281)(132,285)(133,284)(134,283)(135,282)(136,286)
(137,290)(138,289)(139,288)(140,287)(141,316)(142,320)(143,319)(144,318)
(145,317)(146,311)(147,315)(148,314)(149,313)(150,312)(151,306)(152,310)
(153,309)(154,308)(155,307)(156,301)(157,305)(158,304)(159,303)(160,302);;
s1 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 17)( 12, 16)( 13, 20)( 14, 19)
( 15, 18)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 37)( 32, 36)( 33, 40)
( 34, 39)( 35, 38)( 41, 52)( 42, 51)( 43, 55)( 44, 54)( 45, 53)( 46, 57)
( 47, 56)( 48, 60)( 49, 59)( 50, 58)( 61, 72)( 62, 71)( 63, 75)( 64, 74)
( 65, 73)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 81,102)( 82,101)
( 83,105)( 84,104)( 85,103)( 86,107)( 87,106)( 88,110)( 89,109)( 90,108)
( 91,117)( 92,116)( 93,120)( 94,119)( 95,118)( 96,112)( 97,111)( 98,115)
( 99,114)(100,113)(121,152)(122,151)(123,155)(124,154)(125,153)(126,157)
(127,156)(128,160)(129,159)(130,158)(131,142)(132,141)(133,145)(134,144)
(135,143)(136,147)(137,146)(138,150)(139,149)(140,148)(161,202)(162,201)
(163,205)(164,204)(165,203)(166,207)(167,206)(168,210)(169,209)(170,208)
(171,217)(172,216)(173,220)(174,219)(175,218)(176,212)(177,211)(178,215)
(179,214)(180,213)(181,222)(182,221)(183,225)(184,224)(185,223)(186,227)
(187,226)(188,230)(189,229)(190,228)(191,237)(192,236)(193,240)(194,239)
(195,238)(196,232)(197,231)(198,235)(199,234)(200,233)(241,307)(242,306)
(243,310)(244,309)(245,308)(246,302)(247,301)(248,305)(249,304)(250,303)
(251,312)(252,311)(253,315)(254,314)(255,313)(256,317)(257,316)(258,320)
(259,319)(260,318)(261,287)(262,286)(263,290)(264,289)(265,288)(266,282)
(267,281)(268,285)(269,284)(270,283)(271,292)(272,291)(273,295)(274,294)
(275,293)(276,297)(277,296)(278,300)(279,299)(280,298);;
s2 := (  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)
( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,121)( 47,122)( 48,123)
( 49,124)( 50,125)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,131)
( 57,132)( 58,133)( 59,134)( 60,135)( 61,146)( 62,147)( 63,148)( 64,149)
( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)( 72,157)
( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)
(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)
(201,286)(202,287)(203,288)(204,289)(205,290)(206,281)(207,282)(208,283)
(209,284)(210,285)(211,296)(212,297)(213,298)(214,299)(215,300)(216,291)
(217,292)(218,293)(219,294)(220,295)(221,306)(222,307)(223,308)(224,309)
(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)(232,317)
(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)(240,315);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s0*s2*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*s2*s1*s0*s1*s0*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope