Polytope of Type {84,10}
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.
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