Part of the Atlas of Small Regular Polytopes

Polytope of Type {40,20}

Atlas Canonical Name {40,20}*1600e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1600,3549)
Rank
3
Schläfli Type
{40,20}
Vertices, edges, …
40, 400, 20
Order of s0s1s2
40
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

5-fold

8-fold

10-fold

20-fold

25-fold

40-fold

50-fold

80-fold

100-fold

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

Twisty Puzzle