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# Polytope of Type {8,4,10}

Atlas Canonical Name : {8,4,10}*640b
if this polytope has a name.
Group : SmallGroup(640,14089)
Rank : 4
Schlafli Type : {8,4,10}
Number of vertices, edges, etc : 8, 16, 20, 10
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,4,10,2} of size 1280
Vertex Figure Of :
{2,8,4,10} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,10}*320
4-fold quotients : {2,4,10}*160, {4,2,10}*160
5-fold quotients : {8,4,2}*128b
8-fold quotients : {4,2,5}*80, {2,2,10}*80
10-fold quotients : {4,4,2}*64
16-fold quotients : {2,2,5}*40
20-fold quotients : {2,4,2}*32, {4,2,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,4,10}*1280a, {8,8,10}*1280a, {8,8,10}*1280d, {8,4,20}*1280b
3-fold covers : {8,4,30}*1920b, {8,12,10}*1920b, {24,4,10}*1920b
Permutation Representation (GAP) :
```s0 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)
(  9,169)( 10,170)( 11,176)( 12,177)( 13,178)( 14,179)( 15,180)( 16,171)
( 17,172)( 18,173)( 19,174)( 20,175)( 21,186)( 22,187)( 23,188)( 24,189)
( 25,190)( 26,181)( 27,182)( 28,183)( 29,184)( 30,185)( 31,191)( 32,192)
( 33,193)( 34,194)( 35,195)( 36,196)( 37,197)( 38,198)( 39,199)( 40,200)
( 41,201)( 42,202)( 43,203)( 44,204)( 45,205)( 46,206)( 47,207)( 48,208)
( 49,209)( 50,210)( 51,216)( 52,217)( 53,218)( 54,219)( 55,220)( 56,211)
( 57,212)( 58,213)( 59,214)( 60,215)( 61,226)( 62,227)( 63,228)( 64,229)
( 65,230)( 66,221)( 67,222)( 68,223)( 69,224)( 70,225)( 71,231)( 72,232)
( 73,233)( 74,234)( 75,235)( 76,236)( 77,237)( 78,238)( 79,239)( 80,240)
( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)( 88,248)
( 89,249)( 90,250)( 91,256)( 92,257)( 93,258)( 94,259)( 95,260)( 96,251)
( 97,252)( 98,253)( 99,254)(100,255)(101,266)(102,267)(103,268)(104,269)
(105,270)(106,261)(107,262)(108,263)(109,264)(110,265)(111,271)(112,272)
(113,273)(114,274)(115,275)(116,276)(117,277)(118,278)(119,279)(120,280)
(121,281)(122,282)(123,283)(124,284)(125,285)(126,286)(127,287)(128,288)
(129,289)(130,290)(131,296)(132,297)(133,298)(134,299)(135,300)(136,291)
(137,292)(138,293)(139,294)(140,295)(141,306)(142,307)(143,308)(144,309)
(145,310)(146,301)(147,302)(148,303)(149,304)(150,305)(151,311)(152,312)
(153,313)(154,314)(155,315)(156,316)(157,317)(158,318)(159,319)(160,320);;
s1 := ( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 71, 76)
( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81, 91)( 82, 92)( 83, 93)( 84, 94)
( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,116)(102,117)
(103,118)(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)(110,115)
(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)
(129,139)(130,140)(141,156)(142,157)(143,158)(144,159)(145,160)(146,151)
(147,152)(148,153)(149,154)(150,155)(161,181)(162,182)(163,183)(164,184)
(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)
(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)
(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)
(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)
(217,237)(218,238)(219,239)(220,240)(241,276)(242,277)(243,278)(244,279)
(245,280)(246,271)(247,272)(248,273)(249,274)(250,275)(251,266)(252,267)
(253,268)(254,269)(255,270)(256,261)(257,262)(258,263)(259,264)(260,265)
(281,316)(282,317)(283,318)(284,319)(285,320)(286,311)(287,312)(288,313)
(289,314)(290,315)(291,306)(292,307)(293,308)(294,309)(295,310)(296,301)
(297,302)(298,303)(299,304)(300,305);;
s2 := (  1,121)(  2,125)(  3,124)(  4,123)(  5,122)(  6,126)(  7,130)(  8,129)
(  9,128)( 10,127)( 11,131)( 12,135)( 13,134)( 14,133)( 15,132)( 16,136)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,146)( 22,150)( 23,149)( 24,148)
( 25,147)( 26,141)( 27,145)( 28,144)( 29,143)( 30,142)( 31,156)( 32,160)
( 33,159)( 34,158)( 35,157)( 36,151)( 37,155)( 38,154)( 39,153)( 40,152)
( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 46, 86)( 47, 90)( 48, 89)
( 49, 88)( 50, 87)( 51, 91)( 52, 95)( 53, 94)( 54, 93)( 55, 92)( 56, 96)
( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61,106)( 62,110)( 63,109)( 64,108)
( 65,107)( 66,101)( 67,105)( 68,104)( 69,103)( 70,102)( 71,116)( 72,120)
( 73,119)( 74,118)( 75,117)( 76,111)( 77,115)( 78,114)( 79,113)( 80,112)
(161,281)(162,285)(163,284)(164,283)(165,282)(166,286)(167,290)(168,289)
(169,288)(170,287)(171,291)(172,295)(173,294)(174,293)(175,292)(176,296)
(177,300)(178,299)(179,298)(180,297)(181,306)(182,310)(183,309)(184,308)
(185,307)(186,301)(187,305)(188,304)(189,303)(190,302)(191,316)(192,320)
(193,319)(194,318)(195,317)(196,311)(197,315)(198,314)(199,313)(200,312)
(201,241)(202,245)(203,244)(204,243)(205,242)(206,246)(207,250)(208,249)
(209,248)(210,247)(211,251)(212,255)(213,254)(214,253)(215,252)(216,256)
(217,260)(218,259)(219,258)(220,257)(221,266)(222,270)(223,269)(224,268)
(225,267)(226,261)(227,265)(228,264)(229,263)(230,262)(231,276)(232,280)
(233,279)(234,278)(235,277)(236,271)(237,275)(238,274)(239,273)(240,272);;
s3 := (  1, 42)(  2, 41)(  3, 45)(  4, 44)(  5, 43)(  6, 47)(  7, 46)(  8, 50)
(  9, 49)( 10, 48)( 11, 52)( 12, 51)( 13, 55)( 14, 54)( 15, 53)( 16, 57)
( 17, 56)( 18, 60)( 19, 59)( 20, 58)( 21, 62)( 22, 61)( 23, 65)( 24, 64)
( 25, 63)( 26, 67)( 27, 66)( 28, 70)( 29, 69)( 30, 68)( 31, 72)( 32, 71)
( 33, 75)( 34, 74)( 35, 73)( 36, 77)( 37, 76)( 38, 80)( 39, 79)( 40, 78)
( 81,122)( 82,121)( 83,125)( 84,124)( 85,123)( 86,127)( 87,126)( 88,130)
( 89,129)( 90,128)( 91,132)( 92,131)( 93,135)( 94,134)( 95,133)( 96,137)
( 97,136)( 98,140)( 99,139)(100,138)(101,142)(102,141)(103,145)(104,144)
(105,143)(106,147)(107,146)(108,150)(109,149)(110,148)(111,152)(112,151)
(113,155)(114,154)(115,153)(116,157)(117,156)(118,160)(119,159)(120,158)
(161,202)(162,201)(163,205)(164,204)(165,203)(166,207)(167,206)(168,210)
(169,209)(170,208)(171,212)(172,211)(173,215)(174,214)(175,213)(176,217)
(177,216)(178,220)(179,219)(180,218)(181,222)(182,221)(183,225)(184,224)
(185,223)(186,227)(187,226)(188,230)(189,229)(190,228)(191,232)(192,231)
(193,235)(194,234)(195,233)(196,237)(197,236)(198,240)(199,239)(200,238)
(241,282)(242,281)(243,285)(244,284)(245,283)(246,287)(247,286)(248,290)
(249,289)(250,288)(251,292)(252,291)(253,295)(254,294)(255,293)(256,297)
(257,296)(258,300)(259,299)(260,298)(261,302)(262,301)(263,305)(264,304)
(265,303)(266,307)(267,306)(268,310)(269,309)(270,308)(271,312)(272,311)
(273,315)(274,314)(275,313)(276,317)(277,316)(278,320)(279,319)(280,318);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(320)!(  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)
(  8,168)(  9,169)( 10,170)( 11,176)( 12,177)( 13,178)( 14,179)( 15,180)
( 16,171)( 17,172)( 18,173)( 19,174)( 20,175)( 21,186)( 22,187)( 23,188)
( 24,189)( 25,190)( 26,181)( 27,182)( 28,183)( 29,184)( 30,185)( 31,191)
( 32,192)( 33,193)( 34,194)( 35,195)( 36,196)( 37,197)( 38,198)( 39,199)
( 40,200)( 41,201)( 42,202)( 43,203)( 44,204)( 45,205)( 46,206)( 47,207)
( 48,208)( 49,209)( 50,210)( 51,216)( 52,217)( 53,218)( 54,219)( 55,220)
( 56,211)( 57,212)( 58,213)( 59,214)( 60,215)( 61,226)( 62,227)( 63,228)
( 64,229)( 65,230)( 66,221)( 67,222)( 68,223)( 69,224)( 70,225)( 71,231)
( 72,232)( 73,233)( 74,234)( 75,235)( 76,236)( 77,237)( 78,238)( 79,239)
( 80,240)( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)
( 88,248)( 89,249)( 90,250)( 91,256)( 92,257)( 93,258)( 94,259)( 95,260)
( 96,251)( 97,252)( 98,253)( 99,254)(100,255)(101,266)(102,267)(103,268)
(104,269)(105,270)(106,261)(107,262)(108,263)(109,264)(110,265)(111,271)
(112,272)(113,273)(114,274)(115,275)(116,276)(117,277)(118,278)(119,279)
(120,280)(121,281)(122,282)(123,283)(124,284)(125,285)(126,286)(127,287)
(128,288)(129,289)(130,290)(131,296)(132,297)(133,298)(134,299)(135,300)
(136,291)(137,292)(138,293)(139,294)(140,295)(141,306)(142,307)(143,308)
(144,309)(145,310)(146,301)(147,302)(148,303)(149,304)(150,305)(151,311)
(152,312)(153,313)(154,314)(155,315)(156,316)(157,317)(158,318)(159,319)
(160,320);
s1 := Sym(320)!( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)
( 71, 76)( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81, 91)( 82, 92)( 83, 93)
( 84, 94)( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,116)
(102,117)(103,118)(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)
(110,115)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)
(128,138)(129,139)(130,140)(141,156)(142,157)(143,158)(144,159)(145,160)
(146,151)(147,152)(148,153)(149,154)(150,155)(161,181)(162,182)(163,183)
(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)
(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)
(180,200)(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)
(208,228)(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)
(216,236)(217,237)(218,238)(219,239)(220,240)(241,276)(242,277)(243,278)
(244,279)(245,280)(246,271)(247,272)(248,273)(249,274)(250,275)(251,266)
(252,267)(253,268)(254,269)(255,270)(256,261)(257,262)(258,263)(259,264)
(260,265)(281,316)(282,317)(283,318)(284,319)(285,320)(286,311)(287,312)
(288,313)(289,314)(290,315)(291,306)(292,307)(293,308)(294,309)(295,310)
(296,301)(297,302)(298,303)(299,304)(300,305);
s2 := Sym(320)!(  1,121)(  2,125)(  3,124)(  4,123)(  5,122)(  6,126)(  7,130)
(  8,129)(  9,128)( 10,127)( 11,131)( 12,135)( 13,134)( 14,133)( 15,132)
( 16,136)( 17,140)( 18,139)( 19,138)( 20,137)( 21,146)( 22,150)( 23,149)
( 24,148)( 25,147)( 26,141)( 27,145)( 28,144)( 29,143)( 30,142)( 31,156)
( 32,160)( 33,159)( 34,158)( 35,157)( 36,151)( 37,155)( 38,154)( 39,153)
( 40,152)( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 46, 86)( 47, 90)
( 48, 89)( 49, 88)( 50, 87)( 51, 91)( 52, 95)( 53, 94)( 54, 93)( 55, 92)
( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61,106)( 62,110)( 63,109)
( 64,108)( 65,107)( 66,101)( 67,105)( 68,104)( 69,103)( 70,102)( 71,116)
( 72,120)( 73,119)( 74,118)( 75,117)( 76,111)( 77,115)( 78,114)( 79,113)
( 80,112)(161,281)(162,285)(163,284)(164,283)(165,282)(166,286)(167,290)
(168,289)(169,288)(170,287)(171,291)(172,295)(173,294)(174,293)(175,292)
(176,296)(177,300)(178,299)(179,298)(180,297)(181,306)(182,310)(183,309)
(184,308)(185,307)(186,301)(187,305)(188,304)(189,303)(190,302)(191,316)
(192,320)(193,319)(194,318)(195,317)(196,311)(197,315)(198,314)(199,313)
(200,312)(201,241)(202,245)(203,244)(204,243)(205,242)(206,246)(207,250)
(208,249)(209,248)(210,247)(211,251)(212,255)(213,254)(214,253)(215,252)
(216,256)(217,260)(218,259)(219,258)(220,257)(221,266)(222,270)(223,269)
(224,268)(225,267)(226,261)(227,265)(228,264)(229,263)(230,262)(231,276)
(232,280)(233,279)(234,278)(235,277)(236,271)(237,275)(238,274)(239,273)
(240,272);
s3 := Sym(320)!(  1, 42)(  2, 41)(  3, 45)(  4, 44)(  5, 43)(  6, 47)(  7, 46)
(  8, 50)(  9, 49)( 10, 48)( 11, 52)( 12, 51)( 13, 55)( 14, 54)( 15, 53)
( 16, 57)( 17, 56)( 18, 60)( 19, 59)( 20, 58)( 21, 62)( 22, 61)( 23, 65)
( 24, 64)( 25, 63)( 26, 67)( 27, 66)( 28, 70)( 29, 69)( 30, 68)( 31, 72)
( 32, 71)( 33, 75)( 34, 74)( 35, 73)( 36, 77)( 37, 76)( 38, 80)( 39, 79)
( 40, 78)( 81,122)( 82,121)( 83,125)( 84,124)( 85,123)( 86,127)( 87,126)
( 88,130)( 89,129)( 90,128)( 91,132)( 92,131)( 93,135)( 94,134)( 95,133)
( 96,137)( 97,136)( 98,140)( 99,139)(100,138)(101,142)(102,141)(103,145)
(104,144)(105,143)(106,147)(107,146)(108,150)(109,149)(110,148)(111,152)
(112,151)(113,155)(114,154)(115,153)(116,157)(117,156)(118,160)(119,159)
(120,158)(161,202)(162,201)(163,205)(164,204)(165,203)(166,207)(167,206)
(168,210)(169,209)(170,208)(171,212)(172,211)(173,215)(174,214)(175,213)
(176,217)(177,216)(178,220)(179,219)(180,218)(181,222)(182,221)(183,225)
(184,224)(185,223)(186,227)(187,226)(188,230)(189,229)(190,228)(191,232)
(192,231)(193,235)(194,234)(195,233)(196,237)(197,236)(198,240)(199,239)
(200,238)(241,282)(242,281)(243,285)(244,284)(245,283)(246,287)(247,286)
(248,290)(249,289)(250,288)(251,292)(252,291)(253,295)(254,294)(255,293)
(256,297)(257,296)(258,300)(259,299)(260,298)(261,302)(262,301)(263,305)
(264,304)(265,303)(266,307)(267,306)(268,310)(269,309)(270,308)(271,312)
(272,311)(273,315)(274,314)(275,313)(276,317)(277,316)(278,320)(279,319)
(280,318);
poly := sub<Sym(320)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope