Polytope of Type {28,30}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,30}*1680a
Also Known As : {28,30|2}. if this polytope has another name.
Group : SmallGroup(1680,717)
Rank : 3
Schlafli Type : {28,30}
Number of vertices, edges, etc : 28, 420, 30
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 : {14,30}*840
3-fold quotients : {28,10}*560
5-fold quotients : {28,6}*336a
6-fold quotients : {14,10}*280
7-fold quotients : {4,30}*240a
10-fold quotients : {14,6}*168
14-fold quotients : {2,30}*120
15-fold quotients : {28,2}*112
21-fold quotients : {4,10}*80
28-fold quotients : {2,15}*60
30-fold quotients : {14,2}*56
35-fold quotients : {4,6}*48a
42-fold quotients : {2,10}*40
60-fold quotients : {7,2}*28
70-fold quotients : {2,6}*24
84-fold quotients : {2,5}*20
105-fold quotients : {4,2}*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)( 9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(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)(142,147)(143,146)(144,145)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166)
(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)(185,188)
(186,187)(191,196)(192,195)(193,194)(198,203)(199,202)(200,201)(205,210)
(206,209)(207,208)(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)(246,351)(247,357)(248,356)
(249,355)(250,354)(251,353)(252,352)(253,358)(254,364)(255,363)(256,362)
(257,361)(258,360)(259,359)(260,365)(261,371)(262,370)(263,369)(264,368)
(265,367)(266,366)(267,372)(268,378)(269,377)(270,376)(271,375)(272,374)
(273,373)(274,379)(275,385)(276,384)(277,383)(278,382)(279,381)(280,380)
(281,386)(282,392)(283,391)(284,390)(285,389)(286,388)(287,387)(288,393)
(289,399)(290,398)(291,397)(292,396)(293,395)(294,394)(295,400)(296,406)
(297,405)(298,404)(299,403)(300,402)(301,401)(302,407)(303,413)(304,412)
(305,411)(306,410)(307,409)(308,408)(309,414)(310,420)(311,419)(312,418)
(313,417)(314,416)(315,415);;
s1 := ( 1,212)( 2,211)( 3,217)( 4,216)( 5,215)( 6,214)( 7,213)( 8,240)
( 9,239)( 10,245)( 11,244)( 12,243)( 13,242)( 14,241)( 15,233)( 16,232)
( 17,238)( 18,237)( 19,236)( 20,235)( 21,234)( 22,226)( 23,225)( 24,231)
( 25,230)( 26,229)( 27,228)( 28,227)( 29,219)( 30,218)( 31,224)( 32,223)
( 33,222)( 34,221)( 35,220)( 36,282)( 37,281)( 38,287)( 39,286)( 40,285)
( 41,284)( 42,283)( 43,310)( 44,309)( 45,315)( 46,314)( 47,313)( 48,312)
( 49,311)( 50,303)( 51,302)( 52,308)( 53,307)( 54,306)( 55,305)( 56,304)
( 57,296)( 58,295)( 59,301)( 60,300)( 61,299)( 62,298)( 63,297)( 64,289)
( 65,288)( 66,294)( 67,293)( 68,292)( 69,291)( 70,290)( 71,247)( 72,246)
( 73,252)( 74,251)( 75,250)( 76,249)( 77,248)( 78,275)( 79,274)( 80,280)
( 81,279)( 82,278)( 83,277)( 84,276)( 85,268)( 86,267)( 87,273)( 88,272)
( 89,271)( 90,270)( 91,269)( 92,261)( 93,260)( 94,266)( 95,265)( 96,264)
( 97,263)( 98,262)( 99,254)(100,253)(101,259)(102,258)(103,257)(104,256)
(105,255)(106,317)(107,316)(108,322)(109,321)(110,320)(111,319)(112,318)
(113,345)(114,344)(115,350)(116,349)(117,348)(118,347)(119,346)(120,338)
(121,337)(122,343)(123,342)(124,341)(125,340)(126,339)(127,331)(128,330)
(129,336)(130,335)(131,334)(132,333)(133,332)(134,324)(135,323)(136,329)
(137,328)(138,327)(139,326)(140,325)(141,387)(142,386)(143,392)(144,391)
(145,390)(146,389)(147,388)(148,415)(149,414)(150,420)(151,419)(152,418)
(153,417)(154,416)(155,408)(156,407)(157,413)(158,412)(159,411)(160,410)
(161,409)(162,401)(163,400)(164,406)(165,405)(166,404)(167,403)(168,402)
(169,394)(170,393)(171,399)(172,398)(173,397)(174,396)(175,395)(176,352)
(177,351)(178,357)(179,356)(180,355)(181,354)(182,353)(183,380)(184,379)
(185,385)(186,384)(187,383)(188,382)(189,381)(190,373)(191,372)(192,378)
(193,377)(194,376)(195,375)(196,374)(197,366)(198,365)(199,371)(200,370)
(201,369)(202,368)(203,367)(204,359)(205,358)(206,364)(207,363)(208,362)
(209,361)(210,360);;
s2 := ( 1, 43)( 2, 44)( 3, 45)( 4, 46)( 5, 47)( 6, 48)( 7, 49)( 8, 36)
( 9, 37)( 10, 38)( 11, 39)( 12, 40)( 13, 41)( 14, 42)( 15, 64)( 16, 65)
( 17, 66)( 18, 67)( 19, 68)( 20, 69)( 21, 70)( 22, 57)( 23, 58)( 24, 59)
( 25, 60)( 26, 61)( 27, 62)( 28, 63)( 29, 50)( 30, 51)( 31, 52)( 32, 53)
( 33, 54)( 34, 55)( 35, 56)( 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)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)
(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,169)
(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,162)(128,163)
(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)
(137,158)(138,159)(139,160)(140,161)(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)(211,253)(212,254)(213,255)(214,256)(215,257)(216,258)
(217,259)(218,246)(219,247)(220,248)(221,249)(222,250)(223,251)(224,252)
(225,274)(226,275)(227,276)(228,277)(229,278)(230,279)(231,280)(232,267)
(233,268)(234,269)(235,270)(236,271)(237,272)(238,273)(239,260)(240,261)
(241,262)(242,263)(243,264)(244,265)(245,266)(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)(316,358)(317,359)(318,360)(319,361)(320,362)
(321,363)(322,364)(323,351)(324,352)(325,353)(326,354)(327,355)(328,356)
(329,357)(330,379)(331,380)(332,381)(333,382)(334,383)(335,384)(336,385)
(337,372)(338,373)(339,374)(340,375)(341,376)(342,377)(343,378)(344,365)
(345,366)(346,367)(347,368)(348,369)(349,370)(350,371)(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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
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)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(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)(142,147)(143,146)(144,145)
(149,154)(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)
(165,166)(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)
(185,188)(186,187)(191,196)(192,195)(193,194)(198,203)(199,202)(200,201)
(205,210)(206,209)(207,208)(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)(246,351)(247,357)
(248,356)(249,355)(250,354)(251,353)(252,352)(253,358)(254,364)(255,363)
(256,362)(257,361)(258,360)(259,359)(260,365)(261,371)(262,370)(263,369)
(264,368)(265,367)(266,366)(267,372)(268,378)(269,377)(270,376)(271,375)
(272,374)(273,373)(274,379)(275,385)(276,384)(277,383)(278,382)(279,381)
(280,380)(281,386)(282,392)(283,391)(284,390)(285,389)(286,388)(287,387)
(288,393)(289,399)(290,398)(291,397)(292,396)(293,395)(294,394)(295,400)
(296,406)(297,405)(298,404)(299,403)(300,402)(301,401)(302,407)(303,413)
(304,412)(305,411)(306,410)(307,409)(308,408)(309,414)(310,420)(311,419)
(312,418)(313,417)(314,416)(315,415);
s1 := Sym(420)!( 1,212)( 2,211)( 3,217)( 4,216)( 5,215)( 6,214)( 7,213)
( 8,240)( 9,239)( 10,245)( 11,244)( 12,243)( 13,242)( 14,241)( 15,233)
( 16,232)( 17,238)( 18,237)( 19,236)( 20,235)( 21,234)( 22,226)( 23,225)
( 24,231)( 25,230)( 26,229)( 27,228)( 28,227)( 29,219)( 30,218)( 31,224)
( 32,223)( 33,222)( 34,221)( 35,220)( 36,282)( 37,281)( 38,287)( 39,286)
( 40,285)( 41,284)( 42,283)( 43,310)( 44,309)( 45,315)( 46,314)( 47,313)
( 48,312)( 49,311)( 50,303)( 51,302)( 52,308)( 53,307)( 54,306)( 55,305)
( 56,304)( 57,296)( 58,295)( 59,301)( 60,300)( 61,299)( 62,298)( 63,297)
( 64,289)( 65,288)( 66,294)( 67,293)( 68,292)( 69,291)( 70,290)( 71,247)
( 72,246)( 73,252)( 74,251)( 75,250)( 76,249)( 77,248)( 78,275)( 79,274)
( 80,280)( 81,279)( 82,278)( 83,277)( 84,276)( 85,268)( 86,267)( 87,273)
( 88,272)( 89,271)( 90,270)( 91,269)( 92,261)( 93,260)( 94,266)( 95,265)
( 96,264)( 97,263)( 98,262)( 99,254)(100,253)(101,259)(102,258)(103,257)
(104,256)(105,255)(106,317)(107,316)(108,322)(109,321)(110,320)(111,319)
(112,318)(113,345)(114,344)(115,350)(116,349)(117,348)(118,347)(119,346)
(120,338)(121,337)(122,343)(123,342)(124,341)(125,340)(126,339)(127,331)
(128,330)(129,336)(130,335)(131,334)(132,333)(133,332)(134,324)(135,323)
(136,329)(137,328)(138,327)(139,326)(140,325)(141,387)(142,386)(143,392)
(144,391)(145,390)(146,389)(147,388)(148,415)(149,414)(150,420)(151,419)
(152,418)(153,417)(154,416)(155,408)(156,407)(157,413)(158,412)(159,411)
(160,410)(161,409)(162,401)(163,400)(164,406)(165,405)(166,404)(167,403)
(168,402)(169,394)(170,393)(171,399)(172,398)(173,397)(174,396)(175,395)
(176,352)(177,351)(178,357)(179,356)(180,355)(181,354)(182,353)(183,380)
(184,379)(185,385)(186,384)(187,383)(188,382)(189,381)(190,373)(191,372)
(192,378)(193,377)(194,376)(195,375)(196,374)(197,366)(198,365)(199,371)
(200,370)(201,369)(202,368)(203,367)(204,359)(205,358)(206,364)(207,363)
(208,362)(209,361)(210,360);
s2 := Sym(420)!( 1, 43)( 2, 44)( 3, 45)( 4, 46)( 5, 47)( 6, 48)( 7, 49)
( 8, 36)( 9, 37)( 10, 38)( 11, 39)( 12, 40)( 13, 41)( 14, 42)( 15, 64)
( 16, 65)( 17, 66)( 18, 67)( 19, 68)( 20, 69)( 21, 70)( 22, 57)( 23, 58)
( 24, 59)( 25, 60)( 26, 61)( 27, 62)( 28, 63)( 29, 50)( 30, 51)( 31, 52)
( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 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)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)
(112,154)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)
(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,162)
(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)
(136,157)(137,158)(138,159)(139,160)(140,161)(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)(211,253)(212,254)(213,255)(214,256)(215,257)
(216,258)(217,259)(218,246)(219,247)(220,248)(221,249)(222,250)(223,251)
(224,252)(225,274)(226,275)(227,276)(228,277)(229,278)(230,279)(231,280)
(232,267)(233,268)(234,269)(235,270)(236,271)(237,272)(238,273)(239,260)
(240,261)(241,262)(242,263)(243,264)(244,265)(245,266)(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)(316,358)(317,359)(318,360)(319,361)
(320,362)(321,363)(322,364)(323,351)(324,352)(325,353)(326,354)(327,355)
(328,356)(329,357)(330,379)(331,380)(332,381)(333,382)(334,383)(335,384)
(336,385)(337,372)(338,373)(339,374)(340,375)(341,376)(342,377)(343,378)
(344,365)(345,366)(346,367)(347,368)(348,369)(349,370)(350,371)(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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >;
References : None.
to this polytope