Polytope of Type {70,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {70,12}*1680
Also Known As : {70,12|2}. if this polytope has another name.
Group : SmallGroup(1680,797)
Rank : 3
Schlafli Type : {70,12}
Number of vertices, edges, etc : 70, 420, 12
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 : {70,6}*840
3-fold quotients : {70,4}*560
5-fold quotients : {14,12}*336
6-fold quotients : {70,2}*280
7-fold quotients : {10,12}*240
10-fold quotients : {14,6}*168
12-fold quotients : {35,2}*140
14-fold quotients : {10,6}*120
15-fold quotients : {14,4}*112
21-fold quotients : {10,4}*80
30-fold quotients : {14,2}*56
35-fold quotients : {2,12}*48
42-fold quotients : {10,2}*40
60-fold quotients : {7,2}*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 := ( 2, 7)( 3, 6)( 4, 5)( 8, 29)( 9, 35)( 10, 34)( 11, 33)( 12, 32)
( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)
( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)
( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)
( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)( 80,104)
( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)(114,140)
(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)(122,132)
(123,131)(124,130)(125,129)(126,128)(142,147)(143,146)(144,145)(148,169)
(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,162)(156,168)
(157,167)(158,166)(159,165)(160,164)(161,163)(177,182)(178,181)(179,180)
(183,204)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,197)
(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(212,217)(213,216)
(214,215)(218,239)(219,245)(220,244)(221,243)(222,242)(223,241)(224,240)
(225,232)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233)(247,252)
(248,251)(249,250)(253,274)(254,280)(255,279)(256,278)(257,277)(258,276)
(259,275)(260,267)(261,273)(262,272)(263,271)(264,270)(265,269)(266,268)
(282,287)(283,286)(284,285)(288,309)(289,315)(290,314)(291,313)(292,312)
(293,311)(294,310)(295,302)(296,308)(297,307)(298,306)(299,305)(300,304)
(301,303)(317,322)(318,321)(319,320)(323,344)(324,350)(325,349)(326,348)
(327,347)(328,346)(329,345)(330,337)(331,343)(332,342)(333,341)(334,340)
(335,339)(336,338)(352,357)(353,356)(354,355)(358,379)(359,385)(360,384)
(361,383)(362,382)(363,381)(364,380)(365,372)(366,378)(367,377)(368,376)
(369,375)(370,374)(371,373)(387,392)(388,391)(389,390)(393,414)(394,420)
(395,419)(396,418)(397,417)(398,416)(399,415)(400,407)(401,413)(402,412)
(403,411)(404,410)(405,409)(406,408);;
s1 := ( 1, 9)( 2, 8)( 3, 14)( 4, 13)( 5, 12)( 6, 11)( 7, 10)( 15, 30)
( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)( 24, 28)
( 25, 27)( 36, 79)( 37, 78)( 38, 84)( 39, 83)( 40, 82)( 41, 81)( 42, 80)
( 43, 72)( 44, 71)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)( 50,100)
( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)( 57, 93)( 58, 92)
( 59, 98)( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 86)( 65, 85)( 66, 91)
( 67, 90)( 68, 89)( 69, 88)( 70, 87)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)
(125,137)(126,136)(127,128)(129,133)(130,132)(141,184)(142,183)(143,189)
(144,188)(145,187)(146,186)(147,185)(148,177)(149,176)(150,182)(151,181)
(152,180)(153,179)(154,178)(155,205)(156,204)(157,210)(158,209)(159,208)
(160,207)(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)(167,200)
(168,199)(169,191)(170,190)(171,196)(172,195)(173,194)(174,193)(175,192)
(211,324)(212,323)(213,329)(214,328)(215,327)(216,326)(217,325)(218,317)
(219,316)(220,322)(221,321)(222,320)(223,319)(224,318)(225,345)(226,344)
(227,350)(228,349)(229,348)(230,347)(231,346)(232,338)(233,337)(234,343)
(235,342)(236,341)(237,340)(238,339)(239,331)(240,330)(241,336)(242,335)
(243,334)(244,333)(245,332)(246,394)(247,393)(248,399)(249,398)(250,397)
(251,396)(252,395)(253,387)(254,386)(255,392)(256,391)(257,390)(258,389)
(259,388)(260,415)(261,414)(262,420)(263,419)(264,418)(265,417)(266,416)
(267,408)(268,407)(269,413)(270,412)(271,411)(272,410)(273,409)(274,401)
(275,400)(276,406)(277,405)(278,404)(279,403)(280,402)(281,359)(282,358)
(283,364)(284,363)(285,362)(286,361)(287,360)(288,352)(289,351)(290,357)
(291,356)(292,355)(293,354)(294,353)(295,380)(296,379)(297,385)(298,384)
(299,383)(300,382)(301,381)(302,373)(303,372)(304,378)(305,377)(306,376)
(307,375)(308,374)(309,366)(310,365)(311,371)(312,370)(313,369)(314,368)
(315,367);;
s2 := ( 1,246)( 2,247)( 3,248)( 4,249)( 5,250)( 6,251)( 7,252)( 8,253)
( 9,254)( 10,255)( 11,256)( 12,257)( 13,258)( 14,259)( 15,260)( 16,261)
( 17,262)( 18,263)( 19,264)( 20,265)( 21,266)( 22,267)( 23,268)( 24,269)
( 25,270)( 26,271)( 27,272)( 28,273)( 29,274)( 30,275)( 31,276)( 32,277)
( 33,278)( 34,279)( 35,280)( 36,211)( 37,212)( 38,213)( 39,214)( 40,215)
( 41,216)( 42,217)( 43,218)( 44,219)( 45,220)( 46,221)( 47,222)( 48,223)
( 49,224)( 50,225)( 51,226)( 52,227)( 53,228)( 54,229)( 55,230)( 56,231)
( 57,232)( 58,233)( 59,234)( 60,235)( 61,236)( 62,237)( 63,238)( 64,239)
( 65,240)( 66,241)( 67,242)( 68,243)( 69,244)( 70,245)( 71,281)( 72,282)
( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)( 80,290)
( 81,291)( 82,292)( 83,293)( 84,294)( 85,295)( 86,296)( 87,297)( 88,298)
( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)( 96,306)
( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)(104,314)
(105,315)(106,351)(107,352)(108,353)(109,354)(110,355)(111,356)(112,357)
(113,358)(114,359)(115,360)(116,361)(117,362)(118,363)(119,364)(120,365)
(121,366)(122,367)(123,368)(124,369)(125,370)(126,371)(127,372)(128,373)
(129,374)(130,375)(131,376)(132,377)(133,378)(134,379)(135,380)(136,381)
(137,382)(138,383)(139,384)(140,385)(141,316)(142,317)(143,318)(144,319)
(145,320)(146,321)(147,322)(148,323)(149,324)(150,325)(151,326)(152,327)
(153,328)(154,329)(155,330)(156,331)(157,332)(158,333)(159,334)(160,335)
(161,336)(162,337)(163,338)(164,339)(165,340)(166,341)(167,342)(168,343)
(169,344)(170,345)(171,346)(172,347)(173,348)(174,349)(175,350)(176,386)
(177,387)(178,388)(179,389)(180,390)(181,391)(182,392)(183,393)(184,394)
(185,395)(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)(192,402)
(193,403)(194,404)(195,405)(196,406)(197,407)(198,408)(199,409)(200,410)
(201,411)(202,412)(203,413)(204,414)(205,415)(206,416)(207,417)(208,418)
(209,419)(210,420);;
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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(420)!( 2, 7)( 3, 6)( 4, 5)( 8, 29)( 9, 35)( 10, 34)( 11, 33)
( 12, 32)( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)
( 20, 24)( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)
( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)
( 54, 60)( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)
( 80,104)( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)
( 88, 96)( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)
(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)
(122,132)(123,131)(124,130)(125,129)(126,128)(142,147)(143,146)(144,145)
(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,162)
(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(177,182)(178,181)
(179,180)(183,204)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)
(190,197)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(212,217)
(213,216)(214,215)(218,239)(219,245)(220,244)(221,243)(222,242)(223,241)
(224,240)(225,232)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233)
(247,252)(248,251)(249,250)(253,274)(254,280)(255,279)(256,278)(257,277)
(258,276)(259,275)(260,267)(261,273)(262,272)(263,271)(264,270)(265,269)
(266,268)(282,287)(283,286)(284,285)(288,309)(289,315)(290,314)(291,313)
(292,312)(293,311)(294,310)(295,302)(296,308)(297,307)(298,306)(299,305)
(300,304)(301,303)(317,322)(318,321)(319,320)(323,344)(324,350)(325,349)
(326,348)(327,347)(328,346)(329,345)(330,337)(331,343)(332,342)(333,341)
(334,340)(335,339)(336,338)(352,357)(353,356)(354,355)(358,379)(359,385)
(360,384)(361,383)(362,382)(363,381)(364,380)(365,372)(366,378)(367,377)
(368,376)(369,375)(370,374)(371,373)(387,392)(388,391)(389,390)(393,414)
(394,420)(395,419)(396,418)(397,417)(398,416)(399,415)(400,407)(401,413)
(402,412)(403,411)(404,410)(405,409)(406,408);
s1 := Sym(420)!( 1, 9)( 2, 8)( 3, 14)( 4, 13)( 5, 12)( 6, 11)( 7, 10)
( 15, 30)( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)
( 24, 28)( 25, 27)( 36, 79)( 37, 78)( 38, 84)( 39, 83)( 40, 82)( 41, 81)
( 42, 80)( 43, 72)( 44, 71)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)
( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)( 57, 93)
( 58, 92)( 59, 98)( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 86)( 65, 85)
( 66, 91)( 67, 90)( 68, 89)( 69, 88)( 70, 87)(106,114)(107,113)(108,119)
(109,118)(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)
(124,138)(125,137)(126,136)(127,128)(129,133)(130,132)(141,184)(142,183)
(143,189)(144,188)(145,187)(146,186)(147,185)(148,177)(149,176)(150,182)
(151,181)(152,180)(153,179)(154,178)(155,205)(156,204)(157,210)(158,209)
(159,208)(160,207)(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)
(167,200)(168,199)(169,191)(170,190)(171,196)(172,195)(173,194)(174,193)
(175,192)(211,324)(212,323)(213,329)(214,328)(215,327)(216,326)(217,325)
(218,317)(219,316)(220,322)(221,321)(222,320)(223,319)(224,318)(225,345)
(226,344)(227,350)(228,349)(229,348)(230,347)(231,346)(232,338)(233,337)
(234,343)(235,342)(236,341)(237,340)(238,339)(239,331)(240,330)(241,336)
(242,335)(243,334)(244,333)(245,332)(246,394)(247,393)(248,399)(249,398)
(250,397)(251,396)(252,395)(253,387)(254,386)(255,392)(256,391)(257,390)
(258,389)(259,388)(260,415)(261,414)(262,420)(263,419)(264,418)(265,417)
(266,416)(267,408)(268,407)(269,413)(270,412)(271,411)(272,410)(273,409)
(274,401)(275,400)(276,406)(277,405)(278,404)(279,403)(280,402)(281,359)
(282,358)(283,364)(284,363)(285,362)(286,361)(287,360)(288,352)(289,351)
(290,357)(291,356)(292,355)(293,354)(294,353)(295,380)(296,379)(297,385)
(298,384)(299,383)(300,382)(301,381)(302,373)(303,372)(304,378)(305,377)
(306,376)(307,375)(308,374)(309,366)(310,365)(311,371)(312,370)(313,369)
(314,368)(315,367);
s2 := Sym(420)!( 1,246)( 2,247)( 3,248)( 4,249)( 5,250)( 6,251)( 7,252)
( 8,253)( 9,254)( 10,255)( 11,256)( 12,257)( 13,258)( 14,259)( 15,260)
( 16,261)( 17,262)( 18,263)( 19,264)( 20,265)( 21,266)( 22,267)( 23,268)
( 24,269)( 25,270)( 26,271)( 27,272)( 28,273)( 29,274)( 30,275)( 31,276)
( 32,277)( 33,278)( 34,279)( 35,280)( 36,211)( 37,212)( 38,213)( 39,214)
( 40,215)( 41,216)( 42,217)( 43,218)( 44,219)( 45,220)( 46,221)( 47,222)
( 48,223)( 49,224)( 50,225)( 51,226)( 52,227)( 53,228)( 54,229)( 55,230)
( 56,231)( 57,232)( 58,233)( 59,234)( 60,235)( 61,236)( 62,237)( 63,238)
( 64,239)( 65,240)( 66,241)( 67,242)( 68,243)( 69,244)( 70,245)( 71,281)
( 72,282)( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)
( 80,290)( 81,291)( 82,292)( 83,293)( 84,294)( 85,295)( 86,296)( 87,297)
( 88,298)( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)
( 96,306)( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)
(104,314)(105,315)(106,351)(107,352)(108,353)(109,354)(110,355)(111,356)
(112,357)(113,358)(114,359)(115,360)(116,361)(117,362)(118,363)(119,364)
(120,365)(121,366)(122,367)(123,368)(124,369)(125,370)(126,371)(127,372)
(128,373)(129,374)(130,375)(131,376)(132,377)(133,378)(134,379)(135,380)
(136,381)(137,382)(138,383)(139,384)(140,385)(141,316)(142,317)(143,318)
(144,319)(145,320)(146,321)(147,322)(148,323)(149,324)(150,325)(151,326)
(152,327)(153,328)(154,329)(155,330)(156,331)(157,332)(158,333)(159,334)
(160,335)(161,336)(162,337)(163,338)(164,339)(165,340)(166,341)(167,342)
(168,343)(169,344)(170,345)(171,346)(172,347)(173,348)(174,349)(175,350)
(176,386)(177,387)(178,388)(179,389)(180,390)(181,391)(182,392)(183,393)
(184,394)(185,395)(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)
(192,402)(193,403)(194,404)(195,405)(196,406)(197,407)(198,408)(199,409)
(200,410)(201,411)(202,412)(203,413)(204,414)(205,415)(206,416)(207,417)
(208,418)(209,419)(210,420);
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*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 >;
References : None.
to this polytope