Polytope of Type {6,30}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1800c
if this polytope has a name.
Group : SmallGroup(1800,586)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 30, 450, 150
Order of s0s1s2 : 6
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,30}*900
3-fold quotients : {6,10}*600a
6-fold quotients : {6,10}*300
25-fold quotients : {6,6}*72a
75-fold quotients : {2,6}*24, {6,2}*24
150-fold quotients : {2,3}*12, {3,2}*12
225-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 6, 25)( 7, 21)( 8, 22)( 9, 23)( 10, 24)( 11, 19)( 12, 20)( 13, 16)
( 14, 17)( 15, 18)( 31, 50)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 44)
( 37, 45)( 38, 41)( 39, 42)( 40, 43)( 56, 75)( 57, 71)( 58, 72)( 59, 73)
( 60, 74)( 61, 69)( 62, 70)( 63, 66)( 64, 67)( 65, 68)( 76,151)( 77,152)
( 78,153)( 79,154)( 80,155)( 81,175)( 82,171)( 83,172)( 84,173)( 85,174)
( 86,169)( 87,170)( 88,166)( 89,167)( 90,168)( 91,163)( 92,164)( 93,165)
( 94,161)( 95,162)( 96,157)( 97,158)( 98,159)( 99,160)(100,156)(101,176)
(102,177)(103,178)(104,179)(105,180)(106,200)(107,196)(108,197)(109,198)
(110,199)(111,194)(112,195)(113,191)(114,192)(115,193)(116,188)(117,189)
(118,190)(119,186)(120,187)(121,182)(122,183)(123,184)(124,185)(125,181)
(126,201)(127,202)(128,203)(129,204)(130,205)(131,225)(132,221)(133,222)
(134,223)(135,224)(136,219)(137,220)(138,216)(139,217)(140,218)(141,213)
(142,214)(143,215)(144,211)(145,212)(146,207)(147,208)(148,209)(149,210)
(150,206)(231,250)(232,246)(233,247)(234,248)(235,249)(236,244)(237,245)
(238,241)(239,242)(240,243)(256,275)(257,271)(258,272)(259,273)(260,274)
(261,269)(262,270)(263,266)(264,267)(265,268)(281,300)(282,296)(283,297)
(284,298)(285,299)(286,294)(287,295)(288,291)(289,292)(290,293)(301,376)
(302,377)(303,378)(304,379)(305,380)(306,400)(307,396)(308,397)(309,398)
(310,399)(311,394)(312,395)(313,391)(314,392)(315,393)(316,388)(317,389)
(318,390)(319,386)(320,387)(321,382)(322,383)(323,384)(324,385)(325,381)
(326,401)(327,402)(328,403)(329,404)(330,405)(331,425)(332,421)(333,422)
(334,423)(335,424)(336,419)(337,420)(338,416)(339,417)(340,418)(341,413)
(342,414)(343,415)(344,411)(345,412)(346,407)(347,408)(348,409)(349,410)
(350,406)(351,426)(352,427)(353,428)(354,429)(355,430)(356,450)(357,446)
(358,447)(359,448)(360,449)(361,444)(362,445)(363,441)(364,442)(365,443)
(366,438)(367,439)(368,440)(369,436)(370,437)(371,432)(372,433)(373,434)
(374,435)(375,431);;
s1 := ( 1, 76)( 2, 82)( 3, 88)( 4, 94)( 5,100)( 6, 96)( 7, 77)( 8, 83)
( 9, 89)( 10, 95)( 11, 91)( 12, 97)( 13, 78)( 14, 84)( 15, 90)( 16, 86)
( 17, 92)( 18, 98)( 19, 79)( 20, 85)( 21, 81)( 22, 87)( 23, 93)( 24, 99)
( 25, 80)( 26,126)( 27,132)( 28,138)( 29,144)( 30,150)( 31,146)( 32,127)
( 33,133)( 34,139)( 35,145)( 36,141)( 37,147)( 38,128)( 39,134)( 40,140)
( 41,136)( 42,142)( 43,148)( 44,129)( 45,135)( 46,131)( 47,137)( 48,143)
( 49,149)( 50,130)( 51,101)( 52,107)( 53,113)( 54,119)( 55,125)( 56,121)
( 57,102)( 58,108)( 59,114)( 60,120)( 61,116)( 62,122)( 63,103)( 64,109)
( 65,115)( 66,111)( 67,117)( 68,123)( 69,104)( 70,110)( 71,106)( 72,112)
( 73,118)( 74,124)( 75,105)(152,157)(153,163)(154,169)(155,175)(156,171)
(159,164)(160,170)(161,166)(162,172)(168,173)(176,201)(177,207)(178,213)
(179,219)(180,225)(181,221)(182,202)(183,208)(184,214)(185,220)(186,216)
(187,222)(188,203)(189,209)(190,215)(191,211)(192,217)(193,223)(194,204)
(195,210)(196,206)(197,212)(198,218)(199,224)(200,205)(226,301)(227,307)
(228,313)(229,319)(230,325)(231,321)(232,302)(233,308)(234,314)(235,320)
(236,316)(237,322)(238,303)(239,309)(240,315)(241,311)(242,317)(243,323)
(244,304)(245,310)(246,306)(247,312)(248,318)(249,324)(250,305)(251,351)
(252,357)(253,363)(254,369)(255,375)(256,371)(257,352)(258,358)(259,364)
(260,370)(261,366)(262,372)(263,353)(264,359)(265,365)(266,361)(267,367)
(268,373)(269,354)(270,360)(271,356)(272,362)(273,368)(274,374)(275,355)
(276,326)(277,332)(278,338)(279,344)(280,350)(281,346)(282,327)(283,333)
(284,339)(285,345)(286,341)(287,347)(288,328)(289,334)(290,340)(291,336)
(292,342)(293,348)(294,329)(295,335)(296,331)(297,337)(298,343)(299,349)
(300,330)(377,382)(378,388)(379,394)(380,400)(381,396)(384,389)(385,395)
(386,391)(387,397)(393,398)(401,426)(402,432)(403,438)(404,444)(405,450)
(406,446)(407,427)(408,433)(409,439)(410,445)(411,441)(412,447)(413,428)
(414,434)(415,440)(416,436)(417,442)(418,448)(419,429)(420,435)(421,431)
(422,437)(423,443)(424,449)(425,430);;
s2 := ( 1,252)( 2,251)( 3,255)( 4,254)( 5,253)( 6,272)( 7,271)( 8,275)
( 9,274)( 10,273)( 11,267)( 12,266)( 13,270)( 14,269)( 15,268)( 16,262)
( 17,261)( 18,265)( 19,264)( 20,263)( 21,257)( 22,256)( 23,260)( 24,259)
( 25,258)( 26,227)( 27,226)( 28,230)( 29,229)( 30,228)( 31,247)( 32,246)
( 33,250)( 34,249)( 35,248)( 36,242)( 37,241)( 38,245)( 39,244)( 40,243)
( 41,237)( 42,236)( 43,240)( 44,239)( 45,238)( 46,232)( 47,231)( 48,235)
( 49,234)( 50,233)( 51,277)( 52,276)( 53,280)( 54,279)( 55,278)( 56,297)
( 57,296)( 58,300)( 59,299)( 60,298)( 61,292)( 62,291)( 63,295)( 64,294)
( 65,293)( 66,287)( 67,286)( 68,290)( 69,289)( 70,288)( 71,282)( 72,281)
( 73,285)( 74,284)( 75,283)( 76,327)( 77,326)( 78,330)( 79,329)( 80,328)
( 81,347)( 82,346)( 83,350)( 84,349)( 85,348)( 86,342)( 87,341)( 88,345)
( 89,344)( 90,343)( 91,337)( 92,336)( 93,340)( 94,339)( 95,338)( 96,332)
( 97,331)( 98,335)( 99,334)(100,333)(101,302)(102,301)(103,305)(104,304)
(105,303)(106,322)(107,321)(108,325)(109,324)(110,323)(111,317)(112,316)
(113,320)(114,319)(115,318)(116,312)(117,311)(118,315)(119,314)(120,313)
(121,307)(122,306)(123,310)(124,309)(125,308)(126,352)(127,351)(128,355)
(129,354)(130,353)(131,372)(132,371)(133,375)(134,374)(135,373)(136,367)
(137,366)(138,370)(139,369)(140,368)(141,362)(142,361)(143,365)(144,364)
(145,363)(146,357)(147,356)(148,360)(149,359)(150,358)(151,402)(152,401)
(153,405)(154,404)(155,403)(156,422)(157,421)(158,425)(159,424)(160,423)
(161,417)(162,416)(163,420)(164,419)(165,418)(166,412)(167,411)(168,415)
(169,414)(170,413)(171,407)(172,406)(173,410)(174,409)(175,408)(176,377)
(177,376)(178,380)(179,379)(180,378)(181,397)(182,396)(183,400)(184,399)
(185,398)(186,392)(187,391)(188,395)(189,394)(190,393)(191,387)(192,386)
(193,390)(194,389)(195,388)(196,382)(197,381)(198,385)(199,384)(200,383)
(201,427)(202,426)(203,430)(204,429)(205,428)(206,447)(207,446)(208,450)
(209,449)(210,448)(211,442)(212,441)(213,445)(214,444)(215,443)(216,437)
(217,436)(218,440)(219,439)(220,438)(221,432)(222,431)(223,435)(224,434)
(225,433);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(450)!( 6, 25)( 7, 21)( 8, 22)( 9, 23)( 10, 24)( 11, 19)( 12, 20)
( 13, 16)( 14, 17)( 15, 18)( 31, 50)( 32, 46)( 33, 47)( 34, 48)( 35, 49)
( 36, 44)( 37, 45)( 38, 41)( 39, 42)( 40, 43)( 56, 75)( 57, 71)( 58, 72)
( 59, 73)( 60, 74)( 61, 69)( 62, 70)( 63, 66)( 64, 67)( 65, 68)( 76,151)
( 77,152)( 78,153)( 79,154)( 80,155)( 81,175)( 82,171)( 83,172)( 84,173)
( 85,174)( 86,169)( 87,170)( 88,166)( 89,167)( 90,168)( 91,163)( 92,164)
( 93,165)( 94,161)( 95,162)( 96,157)( 97,158)( 98,159)( 99,160)(100,156)
(101,176)(102,177)(103,178)(104,179)(105,180)(106,200)(107,196)(108,197)
(109,198)(110,199)(111,194)(112,195)(113,191)(114,192)(115,193)(116,188)
(117,189)(118,190)(119,186)(120,187)(121,182)(122,183)(123,184)(124,185)
(125,181)(126,201)(127,202)(128,203)(129,204)(130,205)(131,225)(132,221)
(133,222)(134,223)(135,224)(136,219)(137,220)(138,216)(139,217)(140,218)
(141,213)(142,214)(143,215)(144,211)(145,212)(146,207)(147,208)(148,209)
(149,210)(150,206)(231,250)(232,246)(233,247)(234,248)(235,249)(236,244)
(237,245)(238,241)(239,242)(240,243)(256,275)(257,271)(258,272)(259,273)
(260,274)(261,269)(262,270)(263,266)(264,267)(265,268)(281,300)(282,296)
(283,297)(284,298)(285,299)(286,294)(287,295)(288,291)(289,292)(290,293)
(301,376)(302,377)(303,378)(304,379)(305,380)(306,400)(307,396)(308,397)
(309,398)(310,399)(311,394)(312,395)(313,391)(314,392)(315,393)(316,388)
(317,389)(318,390)(319,386)(320,387)(321,382)(322,383)(323,384)(324,385)
(325,381)(326,401)(327,402)(328,403)(329,404)(330,405)(331,425)(332,421)
(333,422)(334,423)(335,424)(336,419)(337,420)(338,416)(339,417)(340,418)
(341,413)(342,414)(343,415)(344,411)(345,412)(346,407)(347,408)(348,409)
(349,410)(350,406)(351,426)(352,427)(353,428)(354,429)(355,430)(356,450)
(357,446)(358,447)(359,448)(360,449)(361,444)(362,445)(363,441)(364,442)
(365,443)(366,438)(367,439)(368,440)(369,436)(370,437)(371,432)(372,433)
(373,434)(374,435)(375,431);
s1 := Sym(450)!( 1, 76)( 2, 82)( 3, 88)( 4, 94)( 5,100)( 6, 96)( 7, 77)
( 8, 83)( 9, 89)( 10, 95)( 11, 91)( 12, 97)( 13, 78)( 14, 84)( 15, 90)
( 16, 86)( 17, 92)( 18, 98)( 19, 79)( 20, 85)( 21, 81)( 22, 87)( 23, 93)
( 24, 99)( 25, 80)( 26,126)( 27,132)( 28,138)( 29,144)( 30,150)( 31,146)
( 32,127)( 33,133)( 34,139)( 35,145)( 36,141)( 37,147)( 38,128)( 39,134)
( 40,140)( 41,136)( 42,142)( 43,148)( 44,129)( 45,135)( 46,131)( 47,137)
( 48,143)( 49,149)( 50,130)( 51,101)( 52,107)( 53,113)( 54,119)( 55,125)
( 56,121)( 57,102)( 58,108)( 59,114)( 60,120)( 61,116)( 62,122)( 63,103)
( 64,109)( 65,115)( 66,111)( 67,117)( 68,123)( 69,104)( 70,110)( 71,106)
( 72,112)( 73,118)( 74,124)( 75,105)(152,157)(153,163)(154,169)(155,175)
(156,171)(159,164)(160,170)(161,166)(162,172)(168,173)(176,201)(177,207)
(178,213)(179,219)(180,225)(181,221)(182,202)(183,208)(184,214)(185,220)
(186,216)(187,222)(188,203)(189,209)(190,215)(191,211)(192,217)(193,223)
(194,204)(195,210)(196,206)(197,212)(198,218)(199,224)(200,205)(226,301)
(227,307)(228,313)(229,319)(230,325)(231,321)(232,302)(233,308)(234,314)
(235,320)(236,316)(237,322)(238,303)(239,309)(240,315)(241,311)(242,317)
(243,323)(244,304)(245,310)(246,306)(247,312)(248,318)(249,324)(250,305)
(251,351)(252,357)(253,363)(254,369)(255,375)(256,371)(257,352)(258,358)
(259,364)(260,370)(261,366)(262,372)(263,353)(264,359)(265,365)(266,361)
(267,367)(268,373)(269,354)(270,360)(271,356)(272,362)(273,368)(274,374)
(275,355)(276,326)(277,332)(278,338)(279,344)(280,350)(281,346)(282,327)
(283,333)(284,339)(285,345)(286,341)(287,347)(288,328)(289,334)(290,340)
(291,336)(292,342)(293,348)(294,329)(295,335)(296,331)(297,337)(298,343)
(299,349)(300,330)(377,382)(378,388)(379,394)(380,400)(381,396)(384,389)
(385,395)(386,391)(387,397)(393,398)(401,426)(402,432)(403,438)(404,444)
(405,450)(406,446)(407,427)(408,433)(409,439)(410,445)(411,441)(412,447)
(413,428)(414,434)(415,440)(416,436)(417,442)(418,448)(419,429)(420,435)
(421,431)(422,437)(423,443)(424,449)(425,430);
s2 := Sym(450)!( 1,252)( 2,251)( 3,255)( 4,254)( 5,253)( 6,272)( 7,271)
( 8,275)( 9,274)( 10,273)( 11,267)( 12,266)( 13,270)( 14,269)( 15,268)
( 16,262)( 17,261)( 18,265)( 19,264)( 20,263)( 21,257)( 22,256)( 23,260)
( 24,259)( 25,258)( 26,227)( 27,226)( 28,230)( 29,229)( 30,228)( 31,247)
( 32,246)( 33,250)( 34,249)( 35,248)( 36,242)( 37,241)( 38,245)( 39,244)
( 40,243)( 41,237)( 42,236)( 43,240)( 44,239)( 45,238)( 46,232)( 47,231)
( 48,235)( 49,234)( 50,233)( 51,277)( 52,276)( 53,280)( 54,279)( 55,278)
( 56,297)( 57,296)( 58,300)( 59,299)( 60,298)( 61,292)( 62,291)( 63,295)
( 64,294)( 65,293)( 66,287)( 67,286)( 68,290)( 69,289)( 70,288)( 71,282)
( 72,281)( 73,285)( 74,284)( 75,283)( 76,327)( 77,326)( 78,330)( 79,329)
( 80,328)( 81,347)( 82,346)( 83,350)( 84,349)( 85,348)( 86,342)( 87,341)
( 88,345)( 89,344)( 90,343)( 91,337)( 92,336)( 93,340)( 94,339)( 95,338)
( 96,332)( 97,331)( 98,335)( 99,334)(100,333)(101,302)(102,301)(103,305)
(104,304)(105,303)(106,322)(107,321)(108,325)(109,324)(110,323)(111,317)
(112,316)(113,320)(114,319)(115,318)(116,312)(117,311)(118,315)(119,314)
(120,313)(121,307)(122,306)(123,310)(124,309)(125,308)(126,352)(127,351)
(128,355)(129,354)(130,353)(131,372)(132,371)(133,375)(134,374)(135,373)
(136,367)(137,366)(138,370)(139,369)(140,368)(141,362)(142,361)(143,365)
(144,364)(145,363)(146,357)(147,356)(148,360)(149,359)(150,358)(151,402)
(152,401)(153,405)(154,404)(155,403)(156,422)(157,421)(158,425)(159,424)
(160,423)(161,417)(162,416)(163,420)(164,419)(165,418)(166,412)(167,411)
(168,415)(169,414)(170,413)(171,407)(172,406)(173,410)(174,409)(175,408)
(176,377)(177,376)(178,380)(179,379)(180,378)(181,397)(182,396)(183,400)
(184,399)(185,398)(186,392)(187,391)(188,395)(189,394)(190,393)(191,387)
(192,386)(193,390)(194,389)(195,388)(196,382)(197,381)(198,385)(199,384)
(200,383)(201,427)(202,426)(203,430)(204,429)(205,428)(206,447)(207,446)
(208,450)(209,449)(210,448)(211,442)(212,441)(213,445)(214,444)(215,443)
(216,437)(217,436)(218,440)(219,439)(220,438)(221,432)(222,431)(223,435)
(224,434)(225,433);
poly := sub<Sym(450)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope