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# Polytope of Type {4,10,8}

Atlas Canonical Name : {4,10,8}*640
Also Known As : {{4,10|2},{10,8|2}}. if this polytope has another name.
Group : SmallGroup(640,13837)
Rank : 4
Schlafli Type : {4,10,8}
Number of vertices, edges, etc : 4, 20, 40, 8
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,10,8,2} of size 1280
Vertex Figure Of :
{2,4,10,8} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,10,4}*320, {2,10,8}*320
4-fold quotients : {2,10,4}*160, {4,10,2}*160
5-fold quotients : {4,2,8}*128
8-fold quotients : {2,10,2}*80
10-fold quotients : {4,2,4}*64, {2,2,8}*64
16-fold quotients : {2,5,2}*40
20-fold quotients : {2,2,4}*32, {4,2,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,10,8}*1280, {4,20,8}*1280a, {4,10,16}*1280
3-fold covers : {4,30,8}*1920a, {12,10,8}*1920, {4,10,24}*1920
Permutation Representation (GAP) :
```s0 := (  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);;
s1 := (  1, 61)(  2, 65)(  3, 64)(  4, 63)(  5, 62)(  6, 66)(  7, 70)(  8, 69)
(  9, 68)( 10, 67)( 11, 71)( 12, 75)( 13, 74)( 14, 73)( 15, 72)( 16, 76)
( 17, 80)( 18, 79)( 19, 78)( 20, 77)( 21, 46)( 22, 50)( 23, 49)( 24, 48)
( 25, 47)( 26, 41)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 31, 56)( 32, 60)
( 33, 59)( 34, 58)( 35, 57)( 36, 51)( 37, 55)( 38, 54)( 39, 53)( 40, 52)
( 81,151)( 82,155)( 83,154)( 84,153)( 85,152)( 86,156)( 87,160)( 88,159)
( 89,158)( 90,157)( 91,141)( 92,145)( 93,144)( 94,143)( 95,142)( 96,146)
( 97,150)( 98,149)( 99,148)(100,147)(101,136)(102,140)(103,139)(104,138)
(105,137)(106,131)(107,135)(108,134)(109,133)(110,132)(111,126)(112,130)
(113,129)(114,128)(115,127)(116,121)(117,125)(118,124)(119,123)(120,122)
(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)
(241,291)(242,295)(243,294)(244,293)(245,292)(246,296)(247,300)(248,299)
(249,298)(250,297)(251,281)(252,285)(253,284)(254,283)(255,282)(256,286)
(257,290)(258,289)(259,288)(260,287)(261,311)(262,315)(263,314)(264,313)
(265,312)(266,316)(267,320)(268,319)(269,318)(270,317)(271,301)(272,305)
(273,304)(274,303)(275,302)(276,306)(277,310)(278,309)(279,308)(280,307);;
s2 := (  1,162)(  2,161)(  3,165)(  4,164)(  5,163)(  6,167)(  7,166)(  8,170)
(  9,169)( 10,168)( 11,172)( 12,171)( 13,175)( 14,174)( 15,173)( 16,177)
( 17,176)( 18,180)( 19,179)( 20,178)( 21,187)( 22,186)( 23,190)( 24,189)
( 25,188)( 26,182)( 27,181)( 28,185)( 29,184)( 30,183)( 31,197)( 32,196)
( 33,200)( 34,199)( 35,198)( 36,192)( 37,191)( 38,195)( 39,194)( 40,193)
( 41,222)( 42,221)( 43,225)( 44,224)( 45,223)( 46,227)( 47,226)( 48,230)
( 49,229)( 50,228)( 51,232)( 52,231)( 53,235)( 54,234)( 55,233)( 56,237)
( 57,236)( 58,240)( 59,239)( 60,238)( 61,202)( 62,201)( 63,205)( 64,204)
( 65,203)( 66,207)( 67,206)( 68,210)( 69,209)( 70,208)( 71,212)( 72,211)
( 73,215)( 74,214)( 75,213)( 76,217)( 77,216)( 78,220)( 79,219)( 80,218)
( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,247)( 87,246)( 88,250)
( 89,249)( 90,248)( 91,252)( 92,251)( 93,255)( 94,254)( 95,253)( 96,257)
( 97,256)( 98,260)( 99,259)(100,258)(101,267)(102,266)(103,270)(104,269)
(105,268)(106,262)(107,261)(108,265)(109,264)(110,263)(111,277)(112,276)
(113,280)(114,279)(115,278)(116,272)(117,271)(118,275)(119,274)(120,273)
(121,302)(122,301)(123,305)(124,304)(125,303)(126,307)(127,306)(128,310)
(129,309)(130,308)(131,312)(132,311)(133,315)(134,314)(135,313)(136,317)
(137,316)(138,320)(139,319)(140,318)(141,282)(142,281)(143,285)(144,284)
(145,283)(146,287)(147,286)(148,290)(149,289)(150,288)(151,292)(152,291)
(153,295)(154,294)(155,293)(156,297)(157,296)(158,300)(159,299)(160,298);;
s3 := (  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 56)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 66)( 22, 67)( 23, 68)( 24, 69)
( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)( 32, 77)
( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)
( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)( 88,128)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)( 96,136)
( 97,137)( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)
(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)
(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155)
(161,201)(162,202)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)
(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)
(177,217)(178,218)(179,219)(180,220)(181,226)(182,227)(183,228)(184,229)
(185,230)(186,221)(187,222)(188,223)(189,224)(190,225)(191,236)(192,237)
(193,238)(194,239)(195,240)(196,231)(197,232)(198,233)(199,234)(200,235)
(241,281)(242,282)(243,283)(244,284)(245,285)(246,286)(247,287)(248,288)
(249,289)(250,290)(251,291)(252,292)(253,293)(254,294)(255,295)(256,296)
(257,297)(258,298)(259,299)(260,300)(261,306)(262,307)(263,308)(264,309)
(265,310)(266,301)(267,302)(268,303)(269,304)(270,305)(271,316)(272,317)
(273,318)(274,319)(275,320)(276,311)(277,312)(278,313)(279,314)(280,315);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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, 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);
s1 := Sym(320)!(  1, 61)(  2, 65)(  3, 64)(  4, 63)(  5, 62)(  6, 66)(  7, 70)
(  8, 69)(  9, 68)( 10, 67)( 11, 71)( 12, 75)( 13, 74)( 14, 73)( 15, 72)
( 16, 76)( 17, 80)( 18, 79)( 19, 78)( 20, 77)( 21, 46)( 22, 50)( 23, 49)
( 24, 48)( 25, 47)( 26, 41)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 31, 56)
( 32, 60)( 33, 59)( 34, 58)( 35, 57)( 36, 51)( 37, 55)( 38, 54)( 39, 53)
( 40, 52)( 81,151)( 82,155)( 83,154)( 84,153)( 85,152)( 86,156)( 87,160)
( 88,159)( 89,158)( 90,157)( 91,141)( 92,145)( 93,144)( 94,143)( 95,142)
( 96,146)( 97,150)( 98,149)( 99,148)(100,147)(101,136)(102,140)(103,139)
(104,138)(105,137)(106,131)(107,135)(108,134)(109,133)(110,132)(111,126)
(112,130)(113,129)(114,128)(115,127)(116,121)(117,125)(118,124)(119,123)
(120,122)(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)(241,291)(242,295)(243,294)(244,293)(245,292)(246,296)(247,300)
(248,299)(249,298)(250,297)(251,281)(252,285)(253,284)(254,283)(255,282)
(256,286)(257,290)(258,289)(259,288)(260,287)(261,311)(262,315)(263,314)
(264,313)(265,312)(266,316)(267,320)(268,319)(269,318)(270,317)(271,301)
(272,305)(273,304)(274,303)(275,302)(276,306)(277,310)(278,309)(279,308)
(280,307);
s2 := Sym(320)!(  1,162)(  2,161)(  3,165)(  4,164)(  5,163)(  6,167)(  7,166)
(  8,170)(  9,169)( 10,168)( 11,172)( 12,171)( 13,175)( 14,174)( 15,173)
( 16,177)( 17,176)( 18,180)( 19,179)( 20,178)( 21,187)( 22,186)( 23,190)
( 24,189)( 25,188)( 26,182)( 27,181)( 28,185)( 29,184)( 30,183)( 31,197)
( 32,196)( 33,200)( 34,199)( 35,198)( 36,192)( 37,191)( 38,195)( 39,194)
( 40,193)( 41,222)( 42,221)( 43,225)( 44,224)( 45,223)( 46,227)( 47,226)
( 48,230)( 49,229)( 50,228)( 51,232)( 52,231)( 53,235)( 54,234)( 55,233)
( 56,237)( 57,236)( 58,240)( 59,239)( 60,238)( 61,202)( 62,201)( 63,205)
( 64,204)( 65,203)( 66,207)( 67,206)( 68,210)( 69,209)( 70,208)( 71,212)
( 72,211)( 73,215)( 74,214)( 75,213)( 76,217)( 77,216)( 78,220)( 79,219)
( 80,218)( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,247)( 87,246)
( 88,250)( 89,249)( 90,248)( 91,252)( 92,251)( 93,255)( 94,254)( 95,253)
( 96,257)( 97,256)( 98,260)( 99,259)(100,258)(101,267)(102,266)(103,270)
(104,269)(105,268)(106,262)(107,261)(108,265)(109,264)(110,263)(111,277)
(112,276)(113,280)(114,279)(115,278)(116,272)(117,271)(118,275)(119,274)
(120,273)(121,302)(122,301)(123,305)(124,304)(125,303)(126,307)(127,306)
(128,310)(129,309)(130,308)(131,312)(132,311)(133,315)(134,314)(135,313)
(136,317)(137,316)(138,320)(139,319)(140,318)(141,282)(142,281)(143,285)
(144,284)(145,283)(146,287)(147,286)(148,290)(149,289)(150,288)(151,292)
(152,291)(153,295)(154,294)(155,293)(156,297)(157,296)(158,300)(159,299)
(160,298);
s3 := Sym(320)!(  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)
( 16, 56)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 66)( 22, 67)( 23, 68)
( 24, 69)( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)
( 32, 77)( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)
( 40, 75)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)
( 88,128)( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)
( 96,136)( 97,137)( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)
(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)
(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)
(120,155)(161,201)(162,202)(163,203)(164,204)(165,205)(166,206)(167,207)
(168,208)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)
(176,216)(177,217)(178,218)(179,219)(180,220)(181,226)(182,227)(183,228)
(184,229)(185,230)(186,221)(187,222)(188,223)(189,224)(190,225)(191,236)
(192,237)(193,238)(194,239)(195,240)(196,231)(197,232)(198,233)(199,234)
(200,235)(241,281)(242,282)(243,283)(244,284)(245,285)(246,286)(247,287)
(248,288)(249,289)(250,290)(251,291)(252,292)(253,293)(254,294)(255,295)
(256,296)(257,297)(258,298)(259,299)(260,300)(261,306)(262,307)(263,308)
(264,309)(265,310)(266,301)(267,302)(268,303)(269,304)(270,305)(271,316)
(272,317)(273,318)(274,319)(275,320)(276,311)(277,312)(278,313)(279,314)
(280,315);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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