Polytope of Type {84,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,10}*1680
Also Known As : {84,10|2}. if this polytope has another name.
Group : SmallGroup(1680,771)
Rank : 3
Schlafli Type : {84,10}
Number of vertices, edges, etc : 84, 420, 10
Order of s0s1s2 : 420
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,10}*840
   3-fold quotients : {28,10}*560
   5-fold quotients : {84,2}*336
   6-fold quotients : {14,10}*280
   7-fold quotients : {12,10}*240
   10-fold quotients : {42,2}*168
   14-fold quotients : {6,10}*120
   15-fold quotients : {28,2}*112
   20-fold quotients : {21,2}*84
   21-fold quotients : {4,10}*80
   30-fold quotients : {14,2}*56
   35-fold quotients : {12,2}*48
   42-fold quotients : {2,10}*40
   60-fold quotients : {7,2}*28
   70-fold quotients : {6,2}*24
   84-fold quotients : {2,5}*20
   105-fold quotients : {4,2}*16
   140-fold quotients : {3,2}*12
   210-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

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

Twisty Puzzle