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# Polytope of Type {8,10}

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

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

```
References : None.
to this polytope