Part of the Atlas of Small Regular Polytopes

Polytope of Type {20,42}

Atlas Canonical Name {20,42}*1680a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1680,770)
Rank
3
Schläfli Type
{20,42}
Vertices, edges, …
20, 420, 42
Order of s0s1s2
420
Order of s0s1s2s1
2
Also known as
{20,42|2}. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

5-fold

6-fold

7-fold

10-fold

14-fold

15-fold

20-fold

21-fold

30-fold

35-fold

42-fold

60-fold

70-fold

84-fold

105-fold

140-fold

210-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP) 
s0 := (  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 57)( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 78, 99)( 79,100)( 80,101)( 81,102)( 82,103)( 83,104)( 84,105)( 85, 92)( 86, 93)( 87, 94)( 88, 95)( 89, 96)( 90, 97)( 91, 98)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,344)(219,345)(220,346)(221,347)(222,348)(223,349)(224,350)(225,337)(226,338)(227,339)(228,340)(229,341)(230,342)(231,343)(232,330)(233,331)(234,332)(235,333)(236,334)(237,335)(238,336)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,351)(247,352)(248,353)(249,354)(250,355)(251,356)(252,357)(253,379)(254,380)(255,381)(256,382)(257,383)(258,384)(259,385)(260,372)(261,373)(262,374)(263,375)(264,376)(265,377)(266,378)(267,365)(268,366)(269,367)(270,368)(271,369)(272,370)(273,371)(274,358)(275,359)(276,360)(277,361)(278,362)(279,363)(280,364)(281,386)(282,387)(283,388)(284,389)(285,390)(286,391)(287,392)(288,414)(289,415)(290,416)(291,417)(292,418)(293,419)(294,420)(295,407)(296,408)(297,409)(298,410)(299,411)(300,412)(301,413)(302,400)(303,401)(304,402)(305,403)(306,404)(307,405)(308,406)(309,393)(310,394)(311,395)(312,396)(313,397)(314,398)(315,399);;
s1 := (  1,218)(  2,224)(  3,223)(  4,222)(  5,221)(  6,220)(  7,219)(  8,211)(  9,217)( 10,216)( 11,215)( 12,214)( 13,213)( 14,212)( 15,239)( 16,245)( 17,244)( 18,243)( 19,242)( 20,241)( 21,240)( 22,232)( 23,238)( 24,237)( 25,236)( 26,235)( 27,234)( 28,233)( 29,225)( 30,231)( 31,230)( 32,229)( 33,228)( 34,227)( 35,226)( 36,288)( 37,294)( 38,293)( 39,292)( 40,291)( 41,290)( 42,289)( 43,281)( 44,287)( 45,286)( 46,285)( 47,284)( 48,283)( 49,282)( 50,309)( 51,315)( 52,314)( 53,313)( 54,312)( 55,311)( 56,310)( 57,302)( 58,308)( 59,307)( 60,306)( 61,305)( 62,304)( 63,303)( 64,295)( 65,301)( 66,300)( 67,299)( 68,298)( 69,297)( 70,296)( 71,253)( 72,259)( 73,258)( 74,257)( 75,256)( 76,255)( 77,254)( 78,246)( 79,252)( 80,251)( 81,250)( 82,249)( 83,248)( 84,247)( 85,274)( 86,280)( 87,279)( 88,278)( 89,277)( 90,276)( 91,275)( 92,267)( 93,273)( 94,272)( 95,271)( 96,270)( 97,269)( 98,268)( 99,260)(100,266)(101,265)(102,264)(103,263)(104,262)(105,261)(106,323)(107,329)(108,328)(109,327)(110,326)(111,325)(112,324)(113,316)(114,322)(115,321)(116,320)(117,319)(118,318)(119,317)(120,344)(121,350)(122,349)(123,348)(124,347)(125,346)(126,345)(127,337)(128,343)(129,342)(130,341)(131,340)(132,339)(133,338)(134,330)(135,336)(136,335)(137,334)(138,333)(139,332)(140,331)(141,393)(142,399)(143,398)(144,397)(145,396)(146,395)(147,394)(148,386)(149,392)(150,391)(151,390)(152,389)(153,388)(154,387)(155,414)(156,420)(157,419)(158,418)(159,417)(160,416)(161,415)(162,407)(163,413)(164,412)(165,411)(166,410)(167,409)(168,408)(169,400)(170,406)(171,405)(172,404)(173,403)(174,402)(175,401)(176,358)(177,364)(178,363)(179,362)(180,361)(181,360)(182,359)(183,351)(184,357)(185,356)(186,355)(187,354)(188,353)(189,352)(190,379)(191,385)(192,384)(193,383)(194,382)(195,381)(196,380)(197,372)(198,378)(199,377)(200,376)(201,375)(202,374)(203,373)(204,365)(205,371)(206,370)(207,369)(208,368)(209,367)(210,366);;
s2 := (  1, 37)(  2, 36)(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 38)(  8, 44)(  9, 43)( 10, 49)( 11, 48)( 12, 47)( 13, 46)( 14, 45)( 15, 51)( 16, 50)( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 52)( 22, 58)( 23, 57)( 24, 63)( 25, 62)( 26, 61)( 27, 60)( 28, 59)( 29, 65)( 30, 64)( 31, 70)( 32, 69)( 33, 68)( 34, 67)( 35, 66)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,142)(107,141)(108,147)(109,146)(110,145)(111,144)(112,143)(113,149)(114,148)(115,154)(116,153)(117,152)(118,151)(119,150)(120,156)(121,155)(122,161)(123,160)(124,159)(125,158)(126,157)(127,163)(128,162)(129,168)(130,167)(131,166)(132,165)(133,164)(134,170)(135,169)(136,175)(137,174)(138,173)(139,172)(140,171)(176,177)(178,182)(179,181)(183,184)(185,189)(186,188)(190,191)(192,196)(193,195)(197,198)(199,203)(200,202)(204,205)(206,210)(207,209)(211,247)(212,246)(213,252)(214,251)(215,250)(216,249)(217,248)(218,254)(219,253)(220,259)(221,258)(222,257)(223,256)(224,255)(225,261)(226,260)(227,266)(228,265)(229,264)(230,263)(231,262)(232,268)(233,267)(234,273)(235,272)(236,271)(237,270)(238,269)(239,275)(240,274)(241,280)(242,279)(243,278)(244,277)(245,276)(281,282)(283,287)(284,286)(288,289)(290,294)(291,293)(295,296)(297,301)(298,300)(302,303)(304,308)(305,307)(309,310)(311,315)(312,314)(316,352)(317,351)(318,357)(319,356)(320,355)(321,354)(322,353)(323,359)(324,358)(325,364)(326,363)(327,362)(328,361)(329,360)(330,366)(331,365)(332,371)(333,370)(334,369)(335,368)(336,367)(337,373)(338,372)(339,378)(340,377)(341,376)(342,375)(343,374)(344,380)(345,379)(346,385)(347,384)(348,383)(349,382)(350,381)(386,387)(388,392)(389,391)(393,394)(395,399)(396,398)(400,401)(402,406)(403,405)(407,408)(409,413)(410,412)(414,415)(416,420)(417,419);;
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*s2*s1*s0*s1*s2*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*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(420)!(  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 57)( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 78, 99)( 79,100)( 80,101)( 81,102)( 82,103)( 83,104)( 84,105)( 85, 92)( 86, 93)( 87, 94)( 88, 95)( 89, 96)( 90, 97)( 91, 98)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,344)(219,345)(220,346)(221,347)(222,348)(223,349)(224,350)(225,337)(226,338)(227,339)(228,340)(229,341)(230,342)(231,343)(232,330)(233,331)(234,332)(235,333)(236,334)(237,335)(238,336)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,351)(247,352)(248,353)(249,354)(250,355)(251,356)(252,357)(253,379)(254,380)(255,381)(256,382)(257,383)(258,384)(259,385)(260,372)(261,373)(262,374)(263,375)(264,376)(265,377)(266,378)(267,365)(268,366)(269,367)(270,368)(271,369)(272,370)(273,371)(274,358)(275,359)(276,360)(277,361)(278,362)(279,363)(280,364)(281,386)(282,387)(283,388)(284,389)(285,390)(286,391)(287,392)(288,414)(289,415)(290,416)(291,417)(292,418)(293,419)(294,420)(295,407)(296,408)(297,409)(298,410)(299,411)(300,412)(301,413)(302,400)(303,401)(304,402)(305,403)(306,404)(307,405)(308,406)(309,393)(310,394)(311,395)(312,396)(313,397)(314,398)(315,399);
s1 := Sym(420)!(  1,218)(  2,224)(  3,223)(  4,222)(  5,221)(  6,220)(  7,219)(  8,211)(  9,217)( 10,216)( 11,215)( 12,214)( 13,213)( 14,212)( 15,239)( 16,245)( 17,244)( 18,243)( 19,242)( 20,241)( 21,240)( 22,232)( 23,238)( 24,237)( 25,236)( 26,235)( 27,234)( 28,233)( 29,225)( 30,231)( 31,230)( 32,229)( 33,228)( 34,227)( 35,226)( 36,288)( 37,294)( 38,293)( 39,292)( 40,291)( 41,290)( 42,289)( 43,281)( 44,287)( 45,286)( 46,285)( 47,284)( 48,283)( 49,282)( 50,309)( 51,315)( 52,314)( 53,313)( 54,312)( 55,311)( 56,310)( 57,302)( 58,308)( 59,307)( 60,306)( 61,305)( 62,304)( 63,303)( 64,295)( 65,301)( 66,300)( 67,299)( 68,298)( 69,297)( 70,296)( 71,253)( 72,259)( 73,258)( 74,257)( 75,256)( 76,255)( 77,254)( 78,246)( 79,252)( 80,251)( 81,250)( 82,249)( 83,248)( 84,247)( 85,274)( 86,280)( 87,279)( 88,278)( 89,277)( 90,276)( 91,275)( 92,267)( 93,273)( 94,272)( 95,271)( 96,270)( 97,269)( 98,268)( 99,260)(100,266)(101,265)(102,264)(103,263)(104,262)(105,261)(106,323)(107,329)(108,328)(109,327)(110,326)(111,325)(112,324)(113,316)(114,322)(115,321)(116,320)(117,319)(118,318)(119,317)(120,344)(121,350)(122,349)(123,348)(124,347)(125,346)(126,345)(127,337)(128,343)(129,342)(130,341)(131,340)(132,339)(133,338)(134,330)(135,336)(136,335)(137,334)(138,333)(139,332)(140,331)(141,393)(142,399)(143,398)(144,397)(145,396)(146,395)(147,394)(148,386)(149,392)(150,391)(151,390)(152,389)(153,388)(154,387)(155,414)(156,420)(157,419)(158,418)(159,417)(160,416)(161,415)(162,407)(163,413)(164,412)(165,411)(166,410)(167,409)(168,408)(169,400)(170,406)(171,405)(172,404)(173,403)(174,402)(175,401)(176,358)(177,364)(178,363)(179,362)(180,361)(181,360)(182,359)(183,351)(184,357)(185,356)(186,355)(187,354)(188,353)(189,352)(190,379)(191,385)(192,384)(193,383)(194,382)(195,381)(196,380)(197,372)(198,378)(199,377)(200,376)(201,375)(202,374)(203,373)(204,365)(205,371)(206,370)(207,369)(208,368)(209,367)(210,366);
s2 := Sym(420)!(  1, 37)(  2, 36)(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 38)(  8, 44)(  9, 43)( 10, 49)( 11, 48)( 12, 47)( 13, 46)( 14, 45)( 15, 51)( 16, 50)( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 52)( 22, 58)( 23, 57)( 24, 63)( 25, 62)( 26, 61)( 27, 60)( 28, 59)( 29, 65)( 30, 64)( 31, 70)( 32, 69)( 33, 68)( 34, 67)( 35, 66)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,142)(107,141)(108,147)(109,146)(110,145)(111,144)(112,143)(113,149)(114,148)(115,154)(116,153)(117,152)(118,151)(119,150)(120,156)(121,155)(122,161)(123,160)(124,159)(125,158)(126,157)(127,163)(128,162)(129,168)(130,167)(131,166)(132,165)(133,164)(134,170)(135,169)(136,175)(137,174)(138,173)(139,172)(140,171)(176,177)(178,182)(179,181)(183,184)(185,189)(186,188)(190,191)(192,196)(193,195)(197,198)(199,203)(200,202)(204,205)(206,210)(207,209)(211,247)(212,246)(213,252)(214,251)(215,250)(216,249)(217,248)(218,254)(219,253)(220,259)(221,258)(222,257)(223,256)(224,255)(225,261)(226,260)(227,266)(228,265)(229,264)(230,263)(231,262)(232,268)(233,267)(234,273)(235,272)(236,271)(237,270)(238,269)(239,275)(240,274)(241,280)(242,279)(243,278)(244,277)(245,276)(281,282)(283,287)(284,286)(288,289)(290,294)(291,293)(295,296)(297,301)(298,300)(302,303)(304,308)(305,307)(309,310)(311,315)(312,314)(316,352)(317,351)(318,357)(319,356)(320,355)(321,354)(322,353)(323,359)(324,358)(325,364)(326,363)(327,362)(328,361)(329,360)(330,366)(331,365)(332,371)(333,370)(334,369)(335,368)(336,367)(337,373)(338,372)(339,378)(340,377)(341,376)(342,375)(343,374)(344,380)(345,379)(346,385)(347,384)(348,383)(349,382)(350,381)(386,387)(388,392)(389,391)(393,394)(395,399)(396,398)(400,401)(402,406)(403,405)(407,408)(409,413)(410,412)(414,415)(416,420)(417,419);
poly := sub<Sym(420)|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*s2*s1*s0*s1*s2*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*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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.

Twisty Puzzle