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# Polytope of Type {4,20}

Atlas Canonical Name : {4,20}*640d
if this polytope has a name.
Group : SmallGroup(640,21465)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 16, 160, 80
Order of s0s1s2 : 10
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,20,2} of size 1280
Vertex Figure Of :
{2,4,20} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,10}*320b
4-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,20}*1280f, {8,20}*1280g, {4,20}*1280d
3-fold covers : {12,20}*1920d, {4,60}*1920i
Permutation Representation (GAP) :
```s0 := (  1,177)(  2,178)(  3,180)(  4,179)(  5,182)(  6,181)(  7,183)(  8,184)
(  9,186)( 10,185)( 11,187)( 12,188)( 13,189)( 14,190)( 15,192)( 16,191)
( 17,162)( 18,161)( 19,163)( 20,164)( 21,165)( 22,166)( 23,168)( 24,167)
( 25,169)( 26,170)( 27,172)( 28,171)( 29,174)( 30,173)( 31,175)( 32,176)
( 33,209)( 34,210)( 35,212)( 36,211)( 37,214)( 38,213)( 39,215)( 40,216)
( 41,218)( 42,217)( 43,219)( 44,220)( 45,221)( 46,222)( 47,224)( 48,223)
( 49,194)( 50,193)( 51,195)( 52,196)( 53,197)( 54,198)( 55,200)( 56,199)
( 57,201)( 58,202)( 59,204)( 60,203)( 61,206)( 62,205)( 63,207)( 64,208)
( 65,241)( 66,242)( 67,244)( 68,243)( 69,246)( 70,245)( 71,247)( 72,248)
( 73,250)( 74,249)( 75,251)( 76,252)( 77,253)( 78,254)( 79,256)( 80,255)
( 81,226)( 82,225)( 83,227)( 84,228)( 85,229)( 86,230)( 87,232)( 88,231)
( 89,233)( 90,234)( 91,236)( 92,235)( 93,238)( 94,237)( 95,239)( 96,240)
( 97,273)( 98,274)( 99,276)(100,275)(101,278)(102,277)(103,279)(104,280)
(105,282)(106,281)(107,283)(108,284)(109,285)(110,286)(111,288)(112,287)
(113,258)(114,257)(115,259)(116,260)(117,261)(118,262)(119,264)(120,263)
(121,265)(122,266)(123,268)(124,267)(125,270)(126,269)(127,271)(128,272)
(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)(136,312)
(137,314)(138,313)(139,315)(140,316)(141,317)(142,318)(143,320)(144,319)
(145,290)(146,289)(147,291)(148,292)(149,293)(150,294)(151,296)(152,295)
(153,297)(154,298)(155,300)(156,299)(157,302)(158,301)(159,303)(160,304);;
s1 := (  1, 33)(  2, 34)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 47)(  8, 48)
(  9, 42)( 10, 41)( 11, 63)( 12, 64)( 13, 50)( 14, 49)( 15, 39)( 16, 40)
( 17, 46)( 18, 45)( 19, 59)( 20, 60)( 21, 54)( 22, 53)( 23, 35)( 24, 36)
( 25, 37)( 26, 38)( 27, 51)( 28, 52)( 29, 61)( 30, 62)( 31, 43)( 32, 44)
( 65,129)( 66,130)( 67,151)( 68,152)( 69,153)( 70,154)( 71,143)( 72,144)
( 73,138)( 74,137)( 75,159)( 76,160)( 77,146)( 78,145)( 79,135)( 80,136)
( 81,142)( 82,141)( 83,155)( 84,156)( 85,150)( 86,149)( 87,131)( 88,132)
( 89,133)( 90,134)( 91,147)( 92,148)( 93,157)( 94,158)( 95,139)( 96,140)
( 99,119)(100,120)(101,121)(102,122)(103,111)(104,112)(105,106)(107,127)
(108,128)(109,114)(110,113)(115,123)(116,124)(117,118)(161,194)(162,193)
(163,216)(164,215)(165,218)(166,217)(167,208)(168,207)(169,201)(170,202)
(171,224)(172,223)(173,209)(174,210)(175,200)(176,199)(177,205)(178,206)
(179,220)(180,219)(181,213)(182,214)(183,196)(184,195)(185,198)(186,197)
(187,212)(188,211)(189,222)(190,221)(191,204)(192,203)(225,290)(226,289)
(227,312)(228,311)(229,314)(230,313)(231,304)(232,303)(233,297)(234,298)
(235,320)(236,319)(237,305)(238,306)(239,296)(240,295)(241,301)(242,302)
(243,316)(244,315)(245,309)(246,310)(247,292)(248,291)(249,294)(250,293)
(251,308)(252,307)(253,318)(254,317)(255,300)(256,299)(257,258)(259,280)
(260,279)(261,282)(262,281)(263,272)(264,271)(267,288)(268,287)(269,273)
(270,274)(275,284)(276,283)(285,286);;
s2 := (  1,289)(  2,290)(  3,308)(  4,307)(  5,309)(  6,310)(  7,296)(  8,295)
(  9,320)( 10,319)( 11,301)( 12,302)( 13,299)( 14,300)( 15,314)( 16,313)
( 17,306)( 18,305)( 19,292)( 20,291)( 21,293)( 22,294)( 23,311)( 24,312)
( 25,304)( 26,303)( 27,318)( 28,317)( 29,316)( 30,315)( 31,298)( 32,297)
( 33,257)( 34,258)( 35,276)( 36,275)( 37,277)( 38,278)( 39,264)( 40,263)
( 41,288)( 42,287)( 43,269)( 44,270)( 45,267)( 46,268)( 47,282)( 48,281)
( 49,274)( 50,273)( 51,260)( 52,259)( 53,261)( 54,262)( 55,279)( 56,280)
( 57,272)( 58,271)( 59,286)( 60,285)( 61,284)( 62,283)( 63,266)( 64,265)
( 65,225)( 66,226)( 67,244)( 68,243)( 69,245)( 70,246)( 71,232)( 72,231)
( 73,256)( 74,255)( 75,237)( 76,238)( 77,235)( 78,236)( 79,250)( 80,249)
( 81,242)( 82,241)( 83,228)( 84,227)( 85,229)( 86,230)( 87,247)( 88,248)
( 89,240)( 90,239)( 91,254)( 92,253)( 93,252)( 94,251)( 95,234)( 96,233)
( 97,193)( 98,194)( 99,212)(100,211)(101,213)(102,214)(103,200)(104,199)
(105,224)(106,223)(107,205)(108,206)(109,203)(110,204)(111,218)(112,217)
(113,210)(114,209)(115,196)(116,195)(117,197)(118,198)(119,215)(120,216)
(121,208)(122,207)(123,222)(124,221)(125,220)(126,219)(127,202)(128,201)
(129,161)(130,162)(131,180)(132,179)(133,181)(134,182)(135,168)(136,167)
(137,192)(138,191)(139,173)(140,174)(141,171)(142,172)(143,186)(144,185)
(145,178)(146,177)(147,164)(148,163)(149,165)(150,166)(151,183)(152,184)
(153,176)(154,175)(155,190)(156,189)(157,188)(158,187)(159,170)(160,169);;
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope