# Polytope of Type {10,8,4}

Atlas Canonical Name : {10,8,4}*640a
Also Known As : {{10,8|2},{8,4|2}}. if this polytope has another name.
Group : SmallGroup(640,13813)
Rank : 4
Schlafli Type : {10,8,4}
Number of vertices, edges, etc : 10, 40, 16, 4
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,8,4,2} of size 1280
Vertex Figure Of :
{2,10,8,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,4,4}*320, {10,8,2}*320
4-fold quotients : {10,2,4}*160, {10,4,2}*160
5-fold quotients : {2,8,4}*128a
8-fold quotients : {5,2,4}*80, {10,2,2}*80
10-fold quotients : {2,4,4}*64, {2,8,2}*64
16-fold quotients : {5,2,2}*40
20-fold quotients : {2,2,4}*32, {2,4,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,8,4}*1280a, {10,8,8}*1280b, {10,8,8}*1280c, {20,8,4}*1280d, {10,16,4}*1280a, {10,16,4}*1280b
3-fold covers : {30,8,4}*1920a, {10,8,12}*1920a, {10,24,4}*1920a
Permutation Representation (GAP) :
```s0 := (  1, 41)(  2, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)(  8, 49)
(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 56)
( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)( 24, 63)
( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)( 32, 75)
( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)
( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)
( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)
( 97,140)( 98,139)( 99,138)(100,137)(101,141)(102,145)(103,144)(104,143)
(105,142)(106,146)(107,150)(108,149)(109,148)(110,147)(111,151)(112,155)
(113,154)(114,153)(115,152)(116,156)(117,160)(118,159)(119,158)(120,157)
(161,201)(162,205)(163,204)(164,203)(165,202)(166,206)(167,210)(168,209)
(169,208)(170,207)(171,211)(172,215)(173,214)(174,213)(175,212)(176,216)
(177,220)(178,219)(179,218)(180,217)(181,221)(182,225)(183,224)(184,223)
(185,222)(186,226)(187,230)(188,229)(189,228)(190,227)(191,231)(192,235)
(193,234)(194,233)(195,232)(196,236)(197,240)(198,239)(199,238)(200,237)
(241,281)(242,285)(243,284)(244,283)(245,282)(246,286)(247,290)(248,289)
(249,288)(250,287)(251,291)(252,295)(253,294)(254,293)(255,292)(256,296)
(257,300)(258,299)(259,298)(260,297)(261,301)(262,305)(263,304)(264,303)
(265,302)(266,306)(267,310)(268,309)(269,308)(270,307)(271,311)(272,315)
(273,314)(274,313)(275,312)(276,316)(277,320)(278,319)(279,318)(280,317);;
s1 := (  1,202)(  2,201)(  3,205)(  4,204)(  5,203)(  6,207)(  7,206)(  8,210)
(  9,209)( 10,208)( 11,212)( 12,211)( 13,215)( 14,214)( 15,213)( 16,217)
( 17,216)( 18,220)( 19,219)( 20,218)( 21,227)( 22,226)( 23,230)( 24,229)
( 25,228)( 26,222)( 27,221)( 28,225)( 29,224)( 30,223)( 31,237)( 32,236)
( 33,240)( 34,239)( 35,238)( 36,232)( 37,231)( 38,235)( 39,234)( 40,233)
( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)( 48,170)
( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)( 56,177)
( 57,176)( 58,180)( 59,179)( 60,178)( 61,187)( 62,186)( 63,190)( 64,189)
( 65,188)( 66,182)( 67,181)( 68,185)( 69,184)( 70,183)( 71,197)( 72,196)
( 73,200)( 74,199)( 75,198)( 76,192)( 77,191)( 78,195)( 79,194)( 80,193)
( 81,282)( 82,281)( 83,285)( 84,284)( 85,283)( 86,287)( 87,286)( 88,290)
( 89,289)( 90,288)( 91,292)( 92,291)( 93,295)( 94,294)( 95,293)( 96,297)
( 97,296)( 98,300)( 99,299)(100,298)(101,307)(102,306)(103,310)(104,309)
(105,308)(106,302)(107,301)(108,305)(109,304)(110,303)(111,317)(112,316)
(113,320)(114,319)(115,318)(116,312)(117,311)(118,315)(119,314)(120,313)
(121,242)(122,241)(123,245)(124,244)(125,243)(126,247)(127,246)(128,250)
(129,249)(130,248)(131,252)(132,251)(133,255)(134,254)(135,253)(136,257)
(137,256)(138,260)(139,259)(140,258)(141,267)(142,266)(143,270)(144,269)
(145,268)(146,262)(147,261)(148,265)(149,264)(150,263)(151,277)(152,276)
(153,280)(154,279)(155,278)(156,272)(157,271)(158,275)(159,274)(160,273);;
s2 := ( 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)(241,271)(242,272)(243,273)(244,274)
(245,275)(246,276)(247,277)(248,278)(249,279)(250,280)(251,261)(252,262)
(253,263)(254,264)(255,265)(256,266)(257,267)(258,268)(259,269)(260,270)
(281,311)(282,312)(283,313)(284,314)(285,315)(286,316)(287,317)(288,318)
(289,319)(290,320)(291,301)(292,302)(293,303)(294,304)(295,305)(296,306)
(297,307)(298,308)(299,309)(300,310);;
s3 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)( 24,104)
( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)
( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)
(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)
(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)
(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)
(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)
(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288)
(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)(216,296)
(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)
(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)
(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope