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# Polytope of Type {20,4,4}

Atlas Canonical Name : {20,4,4}*640
Also Known As : {{20,4|2},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(640,8697)
Rank : 4
Schlafli Type : {20,4,4}
Number of vertices, edges, etc : 20, 40, 8, 4
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{20,4,4,2} of size 1280
Vertex Figure Of :
{2,20,4,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,4,2}*320, {20,2,4}*320, {10,4,4}*320
4-fold quotients : {20,2,2}*160, {10,2,4}*160, {10,4,2}*160
5-fold quotients : {4,4,4}*128
8-fold quotients : {5,2,4}*80, {10,2,2}*80
10-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
16-fold quotients : {5,2,2}*40
20-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,4,8}*1280a, {40,4,4}*1280a, {20,4,8}*1280b, {40,4,4}*1280b, {20,8,4}*1280a, {20,4,4}*1280a, {20,4,4}*1280b, {20,8,4}*1280b, {20,8,4}*1280c, {20,8,4}*1280d
3-fold covers : {60,4,4}*1920, {20,12,4}*1920a, {20,4,12}*1920
Permutation Representation (GAP) :
```s0 := (  1,201)(  2,205)(  3,204)(  4,203)(  5,202)(  6,206)(  7,210)(  8,209)
(  9,208)( 10,207)( 11,211)( 12,215)( 13,214)( 14,213)( 15,212)( 16,216)
( 17,220)( 18,219)( 19,218)( 20,217)( 21,221)( 22,225)( 23,224)( 24,223)
( 25,222)( 26,226)( 27,230)( 28,229)( 29,228)( 30,227)( 31,231)( 32,235)
( 33,234)( 34,233)( 35,232)( 36,236)( 37,240)( 38,239)( 39,238)( 40,237)
( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)( 48,169)
( 49,168)( 50,167)( 51,171)( 52,175)( 53,174)( 54,173)( 55,172)( 56,176)
( 57,180)( 58,179)( 59,178)( 60,177)( 61,181)( 62,185)( 63,184)( 64,183)
( 65,182)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,191)( 72,195)
( 73,194)( 74,193)( 75,192)( 76,196)( 77,200)( 78,199)( 79,198)( 80,197)
( 81,286)( 82,290)( 83,289)( 84,288)( 85,287)( 86,281)( 87,285)( 88,284)
( 89,283)( 90,282)( 91,296)( 92,300)( 93,299)( 94,298)( 95,297)( 96,291)
( 97,295)( 98,294)( 99,293)(100,292)(101,306)(102,310)(103,309)(104,308)
(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,316)(112,320)
(113,319)(114,318)(115,317)(116,311)(117,315)(118,314)(119,313)(120,312)
(121,246)(122,250)(123,249)(124,248)(125,247)(126,241)(127,245)(128,244)
(129,243)(130,242)(131,256)(132,260)(133,259)(134,258)(135,257)(136,251)
(137,255)(138,254)(139,253)(140,252)(141,266)(142,270)(143,269)(144,268)
(145,267)(146,261)(147,265)(148,264)(149,263)(150,262)(151,276)(152,280)
(153,279)(154,278)(155,277)(156,271)(157,275)(158,274)(159,273)(160,272);;
s1 := (  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 87)(  7, 86)(  8, 90)
(  9, 89)( 10, 88)( 11, 92)( 12, 91)( 13, 95)( 14, 94)( 15, 93)( 16, 97)
( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21,102)( 22,101)( 23,105)( 24,104)
( 25,103)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,112)( 32,111)
( 33,115)( 34,114)( 35,113)( 36,117)( 37,116)( 38,120)( 39,119)( 40,118)
( 41,122)( 42,121)( 43,125)( 44,124)( 45,123)( 46,127)( 47,126)( 48,130)
( 49,129)( 50,128)( 51,132)( 52,131)( 53,135)( 54,134)( 55,133)( 56,137)
( 57,136)( 58,140)( 59,139)( 60,138)( 61,142)( 62,141)( 63,145)( 64,144)
( 65,143)( 66,147)( 67,146)( 68,150)( 69,149)( 70,148)( 71,152)( 72,151)
( 73,155)( 74,154)( 75,153)( 76,157)( 77,156)( 78,160)( 79,159)( 80,158)
(161,242)(162,241)(163,245)(164,244)(165,243)(166,247)(167,246)(168,250)
(169,249)(170,248)(171,252)(172,251)(173,255)(174,254)(175,253)(176,257)
(177,256)(178,260)(179,259)(180,258)(181,262)(182,261)(183,265)(184,264)
(185,263)(186,267)(187,266)(188,270)(189,269)(190,268)(191,272)(192,271)
(193,275)(194,274)(195,273)(196,277)(197,276)(198,280)(199,279)(200,278)
(201,282)(202,281)(203,285)(204,284)(205,283)(206,287)(207,286)(208,290)
(209,289)(210,288)(211,292)(212,291)(213,295)(214,294)(215,293)(216,297)
(217,296)(218,300)(219,299)(220,298)(221,302)(222,301)(223,305)(224,304)
(225,303)(226,307)(227,306)(228,310)(229,309)(230,308)(231,312)(232,311)
(233,315)(234,314)(235,313)(236,317)(237,316)(238,320)(239,319)(240,318);;
s2 := ( 41, 61)( 42, 62)( 43, 63)( 44, 64)( 45, 65)( 46, 66)( 47, 67)( 48, 68)
( 49, 69)( 50, 70)( 51, 71)( 52, 72)( 53, 73)( 54, 74)( 55, 75)( 56, 76)
( 57, 77)( 58, 78)( 59, 79)( 60, 80)( 81, 91)( 82, 92)( 83, 93)( 84, 94)
( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,111)(102,112)
(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)
(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)
(129,159)(130,160)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)
(137,147)(138,148)(139,149)(140,150)(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)
(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)(281,291)(282,292)(283,293)(284,294)
(285,295)(286,296)(287,297)(288,298)(289,299)(290,300)(301,311)(302,312)
(303,313)(304,314)(305,315)(306,316)(307,317)(308,318)(309,319)(310,320);;
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, 61)( 22, 62)( 23, 63)( 24, 64)
( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 69)( 30, 70)( 31, 71)( 32, 72)
( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)( 40, 80)
( 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)(101,141)(102,142)(103,143)(104,144)
(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)
(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)
(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)(181,221)(182,222)(183,223)(184,224)
(185,225)(186,226)(187,227)(188,228)(189,229)(190,230)(191,231)(192,232)
(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)
(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)(261,301)(262,302)(263,303)(264,304)
(265,305)(266,306)(267,307)(268,308)(269,309)(270,310)(271,311)(272,312)
(273,313)(274,314)(275,315)(276,316)(277,317)(278,318)(279,319)(280,320);;
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!(  1,201)(  2,205)(  3,204)(  4,203)(  5,202)(  6,206)(  7,210)
(  8,209)(  9,208)( 10,207)( 11,211)( 12,215)( 13,214)( 14,213)( 15,212)
( 16,216)( 17,220)( 18,219)( 19,218)( 20,217)( 21,221)( 22,225)( 23,224)
( 24,223)( 25,222)( 26,226)( 27,230)( 28,229)( 29,228)( 30,227)( 31,231)
( 32,235)( 33,234)( 34,233)( 35,232)( 36,236)( 37,240)( 38,239)( 39,238)
( 40,237)( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)
( 48,169)( 49,168)( 50,167)( 51,171)( 52,175)( 53,174)( 54,173)( 55,172)
( 56,176)( 57,180)( 58,179)( 59,178)( 60,177)( 61,181)( 62,185)( 63,184)
( 64,183)( 65,182)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,191)
( 72,195)( 73,194)( 74,193)( 75,192)( 76,196)( 77,200)( 78,199)( 79,198)
( 80,197)( 81,286)( 82,290)( 83,289)( 84,288)( 85,287)( 86,281)( 87,285)
( 88,284)( 89,283)( 90,282)( 91,296)( 92,300)( 93,299)( 94,298)( 95,297)
( 96,291)( 97,295)( 98,294)( 99,293)(100,292)(101,306)(102,310)(103,309)
(104,308)(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,316)
(112,320)(113,319)(114,318)(115,317)(116,311)(117,315)(118,314)(119,313)
(120,312)(121,246)(122,250)(123,249)(124,248)(125,247)(126,241)(127,245)
(128,244)(129,243)(130,242)(131,256)(132,260)(133,259)(134,258)(135,257)
(136,251)(137,255)(138,254)(139,253)(140,252)(141,266)(142,270)(143,269)
(144,268)(145,267)(146,261)(147,265)(148,264)(149,263)(150,262)(151,276)
(152,280)(153,279)(154,278)(155,277)(156,271)(157,275)(158,274)(159,273)
(160,272);
s1 := Sym(320)!(  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 87)(  7, 86)
(  8, 90)(  9, 89)( 10, 88)( 11, 92)( 12, 91)( 13, 95)( 14, 94)( 15, 93)
( 16, 97)( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21,102)( 22,101)( 23,105)
( 24,104)( 25,103)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,112)
( 32,111)( 33,115)( 34,114)( 35,113)( 36,117)( 37,116)( 38,120)( 39,119)
( 40,118)( 41,122)( 42,121)( 43,125)( 44,124)( 45,123)( 46,127)( 47,126)
( 48,130)( 49,129)( 50,128)( 51,132)( 52,131)( 53,135)( 54,134)( 55,133)
( 56,137)( 57,136)( 58,140)( 59,139)( 60,138)( 61,142)( 62,141)( 63,145)
( 64,144)( 65,143)( 66,147)( 67,146)( 68,150)( 69,149)( 70,148)( 71,152)
( 72,151)( 73,155)( 74,154)( 75,153)( 76,157)( 77,156)( 78,160)( 79,159)
( 80,158)(161,242)(162,241)(163,245)(164,244)(165,243)(166,247)(167,246)
(168,250)(169,249)(170,248)(171,252)(172,251)(173,255)(174,254)(175,253)
(176,257)(177,256)(178,260)(179,259)(180,258)(181,262)(182,261)(183,265)
(184,264)(185,263)(186,267)(187,266)(188,270)(189,269)(190,268)(191,272)
(192,271)(193,275)(194,274)(195,273)(196,277)(197,276)(198,280)(199,279)
(200,278)(201,282)(202,281)(203,285)(204,284)(205,283)(206,287)(207,286)
(208,290)(209,289)(210,288)(211,292)(212,291)(213,295)(214,294)(215,293)
(216,297)(217,296)(218,300)(219,299)(220,298)(221,302)(222,301)(223,305)
(224,304)(225,303)(226,307)(227,306)(228,310)(229,309)(230,308)(231,312)
(232,311)(233,315)(234,314)(235,313)(236,317)(237,316)(238,320)(239,319)
(240,318);
s2 := Sym(320)!( 41, 61)( 42, 62)( 43, 63)( 44, 64)( 45, 65)( 46, 66)( 47, 67)
( 48, 68)( 49, 69)( 50, 70)( 51, 71)( 52, 72)( 53, 73)( 54, 74)( 55, 75)
( 56, 76)( 57, 77)( 58, 78)( 59, 79)( 60, 80)( 81, 91)( 82, 92)( 83, 93)
( 84, 94)( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,111)
(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)
(110,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)
(128,158)(129,159)(130,160)(131,141)(132,142)(133,143)(134,144)(135,145)
(136,146)(137,147)(138,148)(139,149)(140,150)(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)(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)(281,291)(282,292)(283,293)
(284,294)(285,295)(286,296)(287,297)(288,298)(289,299)(290,300)(301,311)
(302,312)(303,313)(304,314)(305,315)(306,316)(307,317)(308,318)(309,319)
(310,320);
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, 61)( 22, 62)( 23, 63)
( 24, 64)( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 69)( 30, 70)( 31, 71)
( 32, 72)( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)
( 40, 80)( 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)(101,141)(102,142)(103,143)
(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)
(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)
(120,160)(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)(181,221)(182,222)(183,223)
(184,224)(185,225)(186,226)(187,227)(188,228)(189,229)(190,230)(191,231)
(192,232)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)
(200,240)(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)(261,301)(262,302)(263,303)
(264,304)(265,305)(266,306)(267,307)(268,308)(269,309)(270,310)(271,311)
(272,312)(273,313)(274,314)(275,315)(276,316)(277,317)(278,318)(279,319)
(280,320);
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*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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