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# Polytope of Type {8,10}

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

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

```
References : None.
to this polytope