# Polytope of Type {20,4}

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

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

```
References : None.
to this polytope