Polytope of Type {10,84}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,84}*1680
Also Known As : {10,84|2}. if this polytope has another name.
Group : SmallGroup(1680,771)
Rank : 3
Schlafli Type : {10,84}
Number of vertices, edges, etc : 10, 420, 84
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 : {10,42}*840
3-fold quotients : {10,28}*560
5-fold quotients : {2,84}*336
6-fold quotients : {10,14}*280
7-fold quotients : {10,12}*240
10-fold quotients : {2,42}*168
14-fold quotients : {10,6}*120
15-fold quotients : {2,28}*112
20-fold quotients : {2,21}*84
21-fold quotients : {10,4}*80
30-fold quotients : {2,14}*56
35-fold quotients : {2,12}*48
42-fold quotients : {10,2}*40
60-fold quotients : {2,7}*28
70-fold quotients : {2,6}*24
84-fold quotients : {5,2}*20
105-fold quotients : {2,4}*16
140-fold quotients : {2,3}*12
210-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
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)(218,239)(219,240)(220,241)(221,242)
(222,243)(223,244)(224,245)(225,232)(226,233)(227,234)(228,235)(229,236)
(230,237)(231,238)(253,274)(254,275)(255,276)(256,277)(257,278)(258,279)
(259,280)(260,267)(261,268)(262,269)(263,270)(264,271)(265,272)(266,273)
(288,309)(289,310)(290,311)(291,312)(292,313)(293,314)(294,315)(295,302)
(296,303)(297,304)(298,305)(299,306)(300,307)(301,308)(323,344)(324,345)
(325,346)(326,347)(327,348)(328,349)(329,350)(330,337)(331,338)(332,339)
(333,340)(334,341)(335,342)(336,343)(358,379)(359,380)(360,381)(361,382)
(362,383)(363,384)(364,385)(365,372)(366,373)(367,374)(368,375)(369,376)
(370,377)(371,378)(393,414)(394,415)(395,416)(396,417)(397,418)(398,419)
(399,420)(400,407)(401,408)(402,409)(403,410)(404,411)(405,412)(406,413);;
s1 := ( 1, 8)( 2, 14)( 3, 13)( 4, 12)( 5, 11)( 6, 10)( 7, 9)( 15, 29)
( 16, 35)( 17, 34)( 18, 33)( 19, 32)( 20, 31)( 21, 30)( 23, 28)( 24, 27)
( 25, 26)( 36, 78)( 37, 84)( 38, 83)( 39, 82)( 40, 81)( 41, 80)( 42, 79)
( 43, 71)( 44, 77)( 45, 76)( 46, 75)( 47, 74)( 48, 73)( 49, 72)( 50, 99)
( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)( 56,100)( 57, 92)( 58, 98)
( 59, 97)( 60, 96)( 61, 95)( 62, 94)( 63, 93)( 64, 85)( 65, 91)( 66, 90)
( 67, 89)( 68, 88)( 69, 87)( 70, 86)(106,113)(107,119)(108,118)(109,117)
(110,116)(111,115)(112,114)(120,134)(121,140)(122,139)(123,138)(124,137)
(125,136)(126,135)(128,133)(129,132)(130,131)(141,183)(142,189)(143,188)
(144,187)(145,186)(146,185)(147,184)(148,176)(149,182)(150,181)(151,180)
(152,179)(153,178)(154,177)(155,204)(156,210)(157,209)(158,208)(159,207)
(160,206)(161,205)(162,197)(163,203)(164,202)(165,201)(166,200)(167,199)
(168,198)(169,190)(170,196)(171,195)(172,194)(173,193)(174,192)(175,191)
(211,323)(212,329)(213,328)(214,327)(215,326)(216,325)(217,324)(218,316)
(219,322)(220,321)(221,320)(222,319)(223,318)(224,317)(225,344)(226,350)
(227,349)(228,348)(229,347)(230,346)(231,345)(232,337)(233,343)(234,342)
(235,341)(236,340)(237,339)(238,338)(239,330)(240,336)(241,335)(242,334)
(243,333)(244,332)(245,331)(246,393)(247,399)(248,398)(249,397)(250,396)
(251,395)(252,394)(253,386)(254,392)(255,391)(256,390)(257,389)(258,388)
(259,387)(260,414)(261,420)(262,419)(263,418)(264,417)(265,416)(266,415)
(267,407)(268,413)(269,412)(270,411)(271,410)(272,409)(273,408)(274,400)
(275,406)(276,405)(277,404)(278,403)(279,402)(280,401)(281,358)(282,364)
(283,363)(284,362)(285,361)(286,360)(287,359)(288,351)(289,357)(290,356)
(291,355)(292,354)(293,353)(294,352)(295,379)(296,385)(297,384)(298,383)
(299,382)(300,381)(301,380)(302,372)(303,378)(304,377)(305,376)(306,375)
(307,374)(308,373)(309,365)(310,371)(311,370)(312,369)(313,368)(314,367)
(315,366);;
s2 := ( 1,247)( 2,246)( 3,252)( 4,251)( 5,250)( 6,249)( 7,248)( 8,254)
( 9,253)( 10,259)( 11,258)( 12,257)( 13,256)( 14,255)( 15,261)( 16,260)
( 17,266)( 18,265)( 19,264)( 20,263)( 21,262)( 22,268)( 23,267)( 24,273)
( 25,272)( 26,271)( 27,270)( 28,269)( 29,275)( 30,274)( 31,280)( 32,279)
( 33,278)( 34,277)( 35,276)( 36,212)( 37,211)( 38,217)( 39,216)( 40,215)
( 41,214)( 42,213)( 43,219)( 44,218)( 45,224)( 46,223)( 47,222)( 48,221)
( 49,220)( 50,226)( 51,225)( 52,231)( 53,230)( 54,229)( 55,228)( 56,227)
( 57,233)( 58,232)( 59,238)( 60,237)( 61,236)( 62,235)( 63,234)( 64,240)
( 65,239)( 66,245)( 67,244)( 68,243)( 69,242)( 70,241)( 71,282)( 72,281)
( 73,287)( 74,286)( 75,285)( 76,284)( 77,283)( 78,289)( 79,288)( 80,294)
( 81,293)( 82,292)( 83,291)( 84,290)( 85,296)( 86,295)( 87,301)( 88,300)
( 89,299)( 90,298)( 91,297)( 92,303)( 93,302)( 94,308)( 95,307)( 96,306)
( 97,305)( 98,304)( 99,310)(100,309)(101,315)(102,314)(103,313)(104,312)
(105,311)(106,352)(107,351)(108,357)(109,356)(110,355)(111,354)(112,353)
(113,359)(114,358)(115,364)(116,363)(117,362)(118,361)(119,360)(120,366)
(121,365)(122,371)(123,370)(124,369)(125,368)(126,367)(127,373)(128,372)
(129,378)(130,377)(131,376)(132,375)(133,374)(134,380)(135,379)(136,385)
(137,384)(138,383)(139,382)(140,381)(141,317)(142,316)(143,322)(144,321)
(145,320)(146,319)(147,318)(148,324)(149,323)(150,329)(151,328)(152,327)
(153,326)(154,325)(155,331)(156,330)(157,336)(158,335)(159,334)(160,333)
(161,332)(162,338)(163,337)(164,343)(165,342)(166,341)(167,340)(168,339)
(169,345)(170,344)(171,350)(172,349)(173,348)(174,347)(175,346)(176,387)
(177,386)(178,392)(179,391)(180,390)(181,389)(182,388)(183,394)(184,393)
(185,399)(186,398)(187,397)(188,396)(189,395)(190,401)(191,400)(192,406)
(193,405)(194,404)(195,403)(196,402)(197,408)(198,407)(199,413)(200,412)
(201,411)(202,410)(203,409)(204,415)(205,414)(206,420)(207,419)(208,418)
(209,417)(210,416);;
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,
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*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)(218,239)(219,240)(220,241)
(221,242)(222,243)(223,244)(224,245)(225,232)(226,233)(227,234)(228,235)
(229,236)(230,237)(231,238)(253,274)(254,275)(255,276)(256,277)(257,278)
(258,279)(259,280)(260,267)(261,268)(262,269)(263,270)(264,271)(265,272)
(266,273)(288,309)(289,310)(290,311)(291,312)(292,313)(293,314)(294,315)
(295,302)(296,303)(297,304)(298,305)(299,306)(300,307)(301,308)(323,344)
(324,345)(325,346)(326,347)(327,348)(328,349)(329,350)(330,337)(331,338)
(332,339)(333,340)(334,341)(335,342)(336,343)(358,379)(359,380)(360,381)
(361,382)(362,383)(363,384)(364,385)(365,372)(366,373)(367,374)(368,375)
(369,376)(370,377)(371,378)(393,414)(394,415)(395,416)(396,417)(397,418)
(398,419)(399,420)(400,407)(401,408)(402,409)(403,410)(404,411)(405,412)
(406,413);
s1 := Sym(420)!( 1, 8)( 2, 14)( 3, 13)( 4, 12)( 5, 11)( 6, 10)( 7, 9)
( 15, 29)( 16, 35)( 17, 34)( 18, 33)( 19, 32)( 20, 31)( 21, 30)( 23, 28)
( 24, 27)( 25, 26)( 36, 78)( 37, 84)( 38, 83)( 39, 82)( 40, 81)( 41, 80)
( 42, 79)( 43, 71)( 44, 77)( 45, 76)( 46, 75)( 47, 74)( 48, 73)( 49, 72)
( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)( 56,100)( 57, 92)
( 58, 98)( 59, 97)( 60, 96)( 61, 95)( 62, 94)( 63, 93)( 64, 85)( 65, 91)
( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)(106,113)(107,119)(108,118)
(109,117)(110,116)(111,115)(112,114)(120,134)(121,140)(122,139)(123,138)
(124,137)(125,136)(126,135)(128,133)(129,132)(130,131)(141,183)(142,189)
(143,188)(144,187)(145,186)(146,185)(147,184)(148,176)(149,182)(150,181)
(151,180)(152,179)(153,178)(154,177)(155,204)(156,210)(157,209)(158,208)
(159,207)(160,206)(161,205)(162,197)(163,203)(164,202)(165,201)(166,200)
(167,199)(168,198)(169,190)(170,196)(171,195)(172,194)(173,193)(174,192)
(175,191)(211,323)(212,329)(213,328)(214,327)(215,326)(216,325)(217,324)
(218,316)(219,322)(220,321)(221,320)(222,319)(223,318)(224,317)(225,344)
(226,350)(227,349)(228,348)(229,347)(230,346)(231,345)(232,337)(233,343)
(234,342)(235,341)(236,340)(237,339)(238,338)(239,330)(240,336)(241,335)
(242,334)(243,333)(244,332)(245,331)(246,393)(247,399)(248,398)(249,397)
(250,396)(251,395)(252,394)(253,386)(254,392)(255,391)(256,390)(257,389)
(258,388)(259,387)(260,414)(261,420)(262,419)(263,418)(264,417)(265,416)
(266,415)(267,407)(268,413)(269,412)(270,411)(271,410)(272,409)(273,408)
(274,400)(275,406)(276,405)(277,404)(278,403)(279,402)(280,401)(281,358)
(282,364)(283,363)(284,362)(285,361)(286,360)(287,359)(288,351)(289,357)
(290,356)(291,355)(292,354)(293,353)(294,352)(295,379)(296,385)(297,384)
(298,383)(299,382)(300,381)(301,380)(302,372)(303,378)(304,377)(305,376)
(306,375)(307,374)(308,373)(309,365)(310,371)(311,370)(312,369)(313,368)
(314,367)(315,366);
s2 := Sym(420)!( 1,247)( 2,246)( 3,252)( 4,251)( 5,250)( 6,249)( 7,248)
( 8,254)( 9,253)( 10,259)( 11,258)( 12,257)( 13,256)( 14,255)( 15,261)
( 16,260)( 17,266)( 18,265)( 19,264)( 20,263)( 21,262)( 22,268)( 23,267)
( 24,273)( 25,272)( 26,271)( 27,270)( 28,269)( 29,275)( 30,274)( 31,280)
( 32,279)( 33,278)( 34,277)( 35,276)( 36,212)( 37,211)( 38,217)( 39,216)
( 40,215)( 41,214)( 42,213)( 43,219)( 44,218)( 45,224)( 46,223)( 47,222)
( 48,221)( 49,220)( 50,226)( 51,225)( 52,231)( 53,230)( 54,229)( 55,228)
( 56,227)( 57,233)( 58,232)( 59,238)( 60,237)( 61,236)( 62,235)( 63,234)
( 64,240)( 65,239)( 66,245)( 67,244)( 68,243)( 69,242)( 70,241)( 71,282)
( 72,281)( 73,287)( 74,286)( 75,285)( 76,284)( 77,283)( 78,289)( 79,288)
( 80,294)( 81,293)( 82,292)( 83,291)( 84,290)( 85,296)( 86,295)( 87,301)
( 88,300)( 89,299)( 90,298)( 91,297)( 92,303)( 93,302)( 94,308)( 95,307)
( 96,306)( 97,305)( 98,304)( 99,310)(100,309)(101,315)(102,314)(103,313)
(104,312)(105,311)(106,352)(107,351)(108,357)(109,356)(110,355)(111,354)
(112,353)(113,359)(114,358)(115,364)(116,363)(117,362)(118,361)(119,360)
(120,366)(121,365)(122,371)(123,370)(124,369)(125,368)(126,367)(127,373)
(128,372)(129,378)(130,377)(131,376)(132,375)(133,374)(134,380)(135,379)
(136,385)(137,384)(138,383)(139,382)(140,381)(141,317)(142,316)(143,322)
(144,321)(145,320)(146,319)(147,318)(148,324)(149,323)(150,329)(151,328)
(152,327)(153,326)(154,325)(155,331)(156,330)(157,336)(158,335)(159,334)
(160,333)(161,332)(162,338)(163,337)(164,343)(165,342)(166,341)(167,340)
(168,339)(169,345)(170,344)(171,350)(172,349)(173,348)(174,347)(175,346)
(176,387)(177,386)(178,392)(179,391)(180,390)(181,389)(182,388)(183,394)
(184,393)(185,399)(186,398)(187,397)(188,396)(189,395)(190,401)(191,400)
(192,406)(193,405)(194,404)(195,403)(196,402)(197,408)(198,407)(199,413)
(200,412)(201,411)(202,410)(203,409)(204,415)(205,414)(206,420)(207,419)
(208,418)(209,417)(210,416);
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,
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*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