# Polytope of Type {10,5}

Atlas Canonical Name : {10,5}*640
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 3
Schlafli Type : {10,5}
Number of vertices, edges, etc : 64, 160, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{10,5,2} of size 1280
Vertex Figure Of :
{2,10,5} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,5}*320, {10,5}*320a, {10,5}*320b
4-fold quotients : {5,5}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,5}*1280, {10,10}*1280a
3-fold covers : {10,15}*1920
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)(163,188)(164,187)
(165,175)(166,176)(167,181)(168,182)(169,178)(170,177)(171,172)(173,192)
(174,191)(183,189)(184,190)(185,186)(193,289)(194,290)(195,316)(196,315)
(197,303)(198,304)(199,309)(200,310)(201,306)(202,305)(203,300)(204,299)
(205,320)(206,319)(207,293)(208,294)(209,298)(210,297)(211,307)(212,308)
(213,295)(214,296)(215,317)(216,318)(217,314)(218,313)(219,292)(220,291)
(221,311)(222,312)(223,302)(224,301)(225,257)(226,258)(227,284)(228,283)
(229,271)(230,272)(231,277)(232,278)(233,274)(234,273)(235,268)(236,267)
(237,288)(238,287)(239,261)(240,262)(241,266)(242,265)(243,275)(244,276)
(245,263)(246,264)(247,285)(248,286)(249,282)(250,281)(251,260)(252,259)
(253,279)(254,280)(255,270)(256,269);;
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 := (  1,180)(  2,179)(  3,170)(  4,169)(  5,189)(  6,190)(  7,168)(  8,167)
(  9,164)( 10,163)( 11,185)( 12,186)( 13,173)( 14,174)( 15,183)( 16,184)
( 17,188)( 18,187)( 19,162)( 20,161)( 21,182)( 22,181)( 23,175)( 24,176)
( 25,171)( 26,172)( 27,178)( 28,177)( 29,165)( 30,166)( 31,191)( 32,192)
( 33,308)( 34,307)( 35,298)( 36,297)( 37,317)( 38,318)( 39,296)( 40,295)
( 41,292)( 42,291)( 43,313)( 44,314)( 45,301)( 46,302)( 47,311)( 48,312)
( 49,316)( 50,315)( 51,290)( 52,289)( 53,310)( 54,309)( 55,303)( 56,304)
( 57,299)( 58,300)( 59,306)( 60,305)( 61,293)( 62,294)( 63,319)( 64,320)
( 65,276)( 66,275)( 67,266)( 68,265)( 69,285)( 70,286)( 71,264)( 72,263)
( 73,260)( 74,259)( 75,281)( 76,282)( 77,269)( 78,270)( 79,279)( 80,280)
( 81,284)( 82,283)( 83,258)( 84,257)( 85,278)( 86,277)( 87,271)( 88,272)
( 89,267)( 90,268)( 91,274)( 92,273)( 93,261)( 94,262)( 95,287)( 96,288)
( 97,244)( 98,243)( 99,234)(100,233)(101,253)(102,254)(103,232)(104,231)
(105,228)(106,227)(107,249)(108,250)(109,237)(110,238)(111,247)(112,248)
(113,252)(114,251)(115,226)(116,225)(117,246)(118,245)(119,239)(120,240)
(121,235)(122,236)(123,242)(124,241)(125,229)(126,230)(127,255)(128,256)
(129,212)(130,211)(131,202)(132,201)(133,221)(134,222)(135,200)(136,199)
(137,196)(138,195)(139,217)(140,218)(141,205)(142,206)(143,215)(144,216)
(145,220)(146,219)(147,194)(148,193)(149,214)(150,213)(151,207)(152,208)
(153,203)(154,204)(155,210)(156,209)(157,197)(158,198)(159,223)(160,224);;
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*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*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*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)(163,188)
(164,187)(165,175)(166,176)(167,181)(168,182)(169,178)(170,177)(171,172)
(173,192)(174,191)(183,189)(184,190)(185,186)(193,289)(194,290)(195,316)
(196,315)(197,303)(198,304)(199,309)(200,310)(201,306)(202,305)(203,300)
(204,299)(205,320)(206,319)(207,293)(208,294)(209,298)(210,297)(211,307)
(212,308)(213,295)(214,296)(215,317)(216,318)(217,314)(218,313)(219,292)
(220,291)(221,311)(222,312)(223,302)(224,301)(225,257)(226,258)(227,284)
(228,283)(229,271)(230,272)(231,277)(232,278)(233,274)(234,273)(235,268)
(236,267)(237,288)(238,287)(239,261)(240,262)(241,266)(242,265)(243,275)
(244,276)(245,263)(246,264)(247,285)(248,286)(249,282)(250,281)(251,260)
(252,259)(253,279)(254,280)(255,270)(256,269);
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)!(  1,180)(  2,179)(  3,170)(  4,169)(  5,189)(  6,190)(  7,168)
(  8,167)(  9,164)( 10,163)( 11,185)( 12,186)( 13,173)( 14,174)( 15,183)
( 16,184)( 17,188)( 18,187)( 19,162)( 20,161)( 21,182)( 22,181)( 23,175)
( 24,176)( 25,171)( 26,172)( 27,178)( 28,177)( 29,165)( 30,166)( 31,191)
( 32,192)( 33,308)( 34,307)( 35,298)( 36,297)( 37,317)( 38,318)( 39,296)
( 40,295)( 41,292)( 42,291)( 43,313)( 44,314)( 45,301)( 46,302)( 47,311)
( 48,312)( 49,316)( 50,315)( 51,290)( 52,289)( 53,310)( 54,309)( 55,303)
( 56,304)( 57,299)( 58,300)( 59,306)( 60,305)( 61,293)( 62,294)( 63,319)
( 64,320)( 65,276)( 66,275)( 67,266)( 68,265)( 69,285)( 70,286)( 71,264)
( 72,263)( 73,260)( 74,259)( 75,281)( 76,282)( 77,269)( 78,270)( 79,279)
( 80,280)( 81,284)( 82,283)( 83,258)( 84,257)( 85,278)( 86,277)( 87,271)
( 88,272)( 89,267)( 90,268)( 91,274)( 92,273)( 93,261)( 94,262)( 95,287)
( 96,288)( 97,244)( 98,243)( 99,234)(100,233)(101,253)(102,254)(103,232)
(104,231)(105,228)(106,227)(107,249)(108,250)(109,237)(110,238)(111,247)
(112,248)(113,252)(114,251)(115,226)(116,225)(117,246)(118,245)(119,239)
(120,240)(121,235)(122,236)(123,242)(124,241)(125,229)(126,230)(127,255)
(128,256)(129,212)(130,211)(131,202)(132,201)(133,221)(134,222)(135,200)
(136,199)(137,196)(138,195)(139,217)(140,218)(141,205)(142,206)(143,215)
(144,216)(145,220)(146,219)(147,194)(148,193)(149,214)(150,213)(151,207)
(152,208)(153,203)(154,204)(155,210)(156,209)(157,197)(158,198)(159,223)
(160,224);
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*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope