Polytope of Type {4,186}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,186}*1488a
Also Known As : {4,186|2}. if this polytope has another name.
Group : SmallGroup(1488,197)
Rank : 3
Schlafli Type : {4,186}
Number of vertices, edges, etc : 4, 372, 186
Order of s0s1s2 : 372
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 : {2,186}*744
3-fold quotients : {4,62}*496
4-fold quotients : {2,93}*372
6-fold quotients : {2,62}*248
12-fold quotients : {2,31}*124
31-fold quotients : {4,6}*48a
62-fold quotients : {2,6}*24
93-fold quotients : {4,2}*16
124-fold quotients : {2,3}*12
186-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (187,280)(188,281)(189,282)(190,283)(191,284)(192,285)(193,286)(194,287)
(195,288)(196,289)(197,290)(198,291)(199,292)(200,293)(201,294)(202,295)
(203,296)(204,297)(205,298)(206,299)(207,300)(208,301)(209,302)(210,303)
(211,304)(212,305)(213,306)(214,307)(215,308)(216,309)(217,310)(218,311)
(219,312)(220,313)(221,314)(222,315)(223,316)(224,317)(225,318)(226,319)
(227,320)(228,321)(229,322)(230,323)(231,324)(232,325)(233,326)(234,327)
(235,328)(236,329)(237,330)(238,331)(239,332)(240,333)(241,334)(242,335)
(243,336)(244,337)(245,338)(246,339)(247,340)(248,341)(249,342)(250,343)
(251,344)(252,345)(253,346)(254,347)(255,348)(256,349)(257,350)(258,351)
(259,352)(260,353)(261,354)(262,355)(263,356)(264,357)(265,358)(266,359)
(267,360)(268,361)(269,362)(270,363)(271,364)(272,365)(273,366)(274,367)
(275,368)(276,369)(277,370)(278,371)(279,372);;
s1 := ( 1,187)( 2,217)( 3,216)( 4,215)( 5,214)( 6,213)( 7,212)( 8,211)
( 9,210)( 10,209)( 11,208)( 12,207)( 13,206)( 14,205)( 15,204)( 16,203)
( 17,202)( 18,201)( 19,200)( 20,199)( 21,198)( 22,197)( 23,196)( 24,195)
( 25,194)( 26,193)( 27,192)( 28,191)( 29,190)( 30,189)( 31,188)( 32,249)
( 33,279)( 34,278)( 35,277)( 36,276)( 37,275)( 38,274)( 39,273)( 40,272)
( 41,271)( 42,270)( 43,269)( 44,268)( 45,267)( 46,266)( 47,265)( 48,264)
( 49,263)( 50,262)( 51,261)( 52,260)( 53,259)( 54,258)( 55,257)( 56,256)
( 57,255)( 58,254)( 59,253)( 60,252)( 61,251)( 62,250)( 63,218)( 64,248)
( 65,247)( 66,246)( 67,245)( 68,244)( 69,243)( 70,242)( 71,241)( 72,240)
( 73,239)( 74,238)( 75,237)( 76,236)( 77,235)( 78,234)( 79,233)( 80,232)
( 81,231)( 82,230)( 83,229)( 84,228)( 85,227)( 86,226)( 87,225)( 88,224)
( 89,223)( 90,222)( 91,221)( 92,220)( 93,219)( 94,280)( 95,310)( 96,309)
( 97,308)( 98,307)( 99,306)(100,305)(101,304)(102,303)(103,302)(104,301)
(105,300)(106,299)(107,298)(108,297)(109,296)(110,295)(111,294)(112,293)
(113,292)(114,291)(115,290)(116,289)(117,288)(118,287)(119,286)(120,285)
(121,284)(122,283)(123,282)(124,281)(125,342)(126,372)(127,371)(128,370)
(129,369)(130,368)(131,367)(132,366)(133,365)(134,364)(135,363)(136,362)
(137,361)(138,360)(139,359)(140,358)(141,357)(142,356)(143,355)(144,354)
(145,353)(146,352)(147,351)(148,350)(149,349)(150,348)(151,347)(152,346)
(153,345)(154,344)(155,343)(156,311)(157,341)(158,340)(159,339)(160,338)
(161,337)(162,336)(163,335)(164,334)(165,333)(166,332)(167,331)(168,330)
(169,329)(170,328)(171,327)(172,326)(173,325)(174,324)(175,323)(176,322)
(177,321)(178,320)(179,319)(180,318)(181,317)(182,316)(183,315)(184,314)
(185,313)(186,312);;
s2 := ( 1, 33)( 2, 32)( 3, 62)( 4, 61)( 5, 60)( 6, 59)( 7, 58)( 8, 57)
( 9, 56)( 10, 55)( 11, 54)( 12, 53)( 13, 52)( 14, 51)( 15, 50)( 16, 49)
( 17, 48)( 18, 47)( 19, 46)( 20, 45)( 21, 44)( 22, 43)( 23, 42)( 24, 41)
( 25, 40)( 26, 39)( 27, 38)( 28, 37)( 29, 36)( 30, 35)( 31, 34)( 63, 64)
( 65, 93)( 66, 92)( 67, 91)( 68, 90)( 69, 89)( 70, 88)( 71, 87)( 72, 86)
( 73, 85)( 74, 84)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 94,126)( 95,125)
( 96,155)( 97,154)( 98,153)( 99,152)(100,151)(101,150)(102,149)(103,148)
(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)
(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)
(120,131)(121,130)(122,129)(123,128)(124,127)(156,157)(158,186)(159,185)
(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)
(168,176)(169,175)(170,174)(171,173)(187,219)(188,218)(189,248)(190,247)
(191,246)(192,245)(193,244)(194,243)(195,242)(196,241)(197,240)(198,239)
(199,238)(200,237)(201,236)(202,235)(203,234)(204,233)(205,232)(206,231)
(207,230)(208,229)(209,228)(210,227)(211,226)(212,225)(213,224)(214,223)
(215,222)(216,221)(217,220)(249,250)(251,279)(252,278)(253,277)(254,276)
(255,275)(256,274)(257,273)(258,272)(259,271)(260,270)(261,269)(262,268)
(263,267)(264,266)(280,312)(281,311)(282,341)(283,340)(284,339)(285,338)
(286,337)(287,336)(288,335)(289,334)(290,333)(291,332)(292,331)(293,330)
(294,329)(295,328)(296,327)(297,326)(298,325)(299,324)(300,323)(301,322)
(302,321)(303,320)(304,319)(305,318)(306,317)(307,316)(308,315)(309,314)
(310,313)(342,343)(344,372)(345,371)(346,370)(347,369)(348,368)(349,367)
(350,366)(351,365)(352,364)(353,363)(354,362)(355,361)(356,360)(357,359);;
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*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(372)!(187,280)(188,281)(189,282)(190,283)(191,284)(192,285)(193,286)
(194,287)(195,288)(196,289)(197,290)(198,291)(199,292)(200,293)(201,294)
(202,295)(203,296)(204,297)(205,298)(206,299)(207,300)(208,301)(209,302)
(210,303)(211,304)(212,305)(213,306)(214,307)(215,308)(216,309)(217,310)
(218,311)(219,312)(220,313)(221,314)(222,315)(223,316)(224,317)(225,318)
(226,319)(227,320)(228,321)(229,322)(230,323)(231,324)(232,325)(233,326)
(234,327)(235,328)(236,329)(237,330)(238,331)(239,332)(240,333)(241,334)
(242,335)(243,336)(244,337)(245,338)(246,339)(247,340)(248,341)(249,342)
(250,343)(251,344)(252,345)(253,346)(254,347)(255,348)(256,349)(257,350)
(258,351)(259,352)(260,353)(261,354)(262,355)(263,356)(264,357)(265,358)
(266,359)(267,360)(268,361)(269,362)(270,363)(271,364)(272,365)(273,366)
(274,367)(275,368)(276,369)(277,370)(278,371)(279,372);
s1 := Sym(372)!( 1,187)( 2,217)( 3,216)( 4,215)( 5,214)( 6,213)( 7,212)
( 8,211)( 9,210)( 10,209)( 11,208)( 12,207)( 13,206)( 14,205)( 15,204)
( 16,203)( 17,202)( 18,201)( 19,200)( 20,199)( 21,198)( 22,197)( 23,196)
( 24,195)( 25,194)( 26,193)( 27,192)( 28,191)( 29,190)( 30,189)( 31,188)
( 32,249)( 33,279)( 34,278)( 35,277)( 36,276)( 37,275)( 38,274)( 39,273)
( 40,272)( 41,271)( 42,270)( 43,269)( 44,268)( 45,267)( 46,266)( 47,265)
( 48,264)( 49,263)( 50,262)( 51,261)( 52,260)( 53,259)( 54,258)( 55,257)
( 56,256)( 57,255)( 58,254)( 59,253)( 60,252)( 61,251)( 62,250)( 63,218)
( 64,248)( 65,247)( 66,246)( 67,245)( 68,244)( 69,243)( 70,242)( 71,241)
( 72,240)( 73,239)( 74,238)( 75,237)( 76,236)( 77,235)( 78,234)( 79,233)
( 80,232)( 81,231)( 82,230)( 83,229)( 84,228)( 85,227)( 86,226)( 87,225)
( 88,224)( 89,223)( 90,222)( 91,221)( 92,220)( 93,219)( 94,280)( 95,310)
( 96,309)( 97,308)( 98,307)( 99,306)(100,305)(101,304)(102,303)(103,302)
(104,301)(105,300)(106,299)(107,298)(108,297)(109,296)(110,295)(111,294)
(112,293)(113,292)(114,291)(115,290)(116,289)(117,288)(118,287)(119,286)
(120,285)(121,284)(122,283)(123,282)(124,281)(125,342)(126,372)(127,371)
(128,370)(129,369)(130,368)(131,367)(132,366)(133,365)(134,364)(135,363)
(136,362)(137,361)(138,360)(139,359)(140,358)(141,357)(142,356)(143,355)
(144,354)(145,353)(146,352)(147,351)(148,350)(149,349)(150,348)(151,347)
(152,346)(153,345)(154,344)(155,343)(156,311)(157,341)(158,340)(159,339)
(160,338)(161,337)(162,336)(163,335)(164,334)(165,333)(166,332)(167,331)
(168,330)(169,329)(170,328)(171,327)(172,326)(173,325)(174,324)(175,323)
(176,322)(177,321)(178,320)(179,319)(180,318)(181,317)(182,316)(183,315)
(184,314)(185,313)(186,312);
s2 := Sym(372)!( 1, 33)( 2, 32)( 3, 62)( 4, 61)( 5, 60)( 6, 59)( 7, 58)
( 8, 57)( 9, 56)( 10, 55)( 11, 54)( 12, 53)( 13, 52)( 14, 51)( 15, 50)
( 16, 49)( 17, 48)( 18, 47)( 19, 46)( 20, 45)( 21, 44)( 22, 43)( 23, 42)
( 24, 41)( 25, 40)( 26, 39)( 27, 38)( 28, 37)( 29, 36)( 30, 35)( 31, 34)
( 63, 64)( 65, 93)( 66, 92)( 67, 91)( 68, 90)( 69, 89)( 70, 88)( 71, 87)
( 72, 86)( 73, 85)( 74, 84)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 94,126)
( 95,125)( 96,155)( 97,154)( 98,153)( 99,152)(100,151)(101,150)(102,149)
(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)
(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)
(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(156,157)(158,186)
(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)
(167,177)(168,176)(169,175)(170,174)(171,173)(187,219)(188,218)(189,248)
(190,247)(191,246)(192,245)(193,244)(194,243)(195,242)(196,241)(197,240)
(198,239)(199,238)(200,237)(201,236)(202,235)(203,234)(204,233)(205,232)
(206,231)(207,230)(208,229)(209,228)(210,227)(211,226)(212,225)(213,224)
(214,223)(215,222)(216,221)(217,220)(249,250)(251,279)(252,278)(253,277)
(254,276)(255,275)(256,274)(257,273)(258,272)(259,271)(260,270)(261,269)
(262,268)(263,267)(264,266)(280,312)(281,311)(282,341)(283,340)(284,339)
(285,338)(286,337)(287,336)(288,335)(289,334)(290,333)(291,332)(292,331)
(293,330)(294,329)(295,328)(296,327)(297,326)(298,325)(299,324)(300,323)
(301,322)(302,321)(303,320)(304,319)(305,318)(306,317)(307,316)(308,315)
(309,314)(310,313)(342,343)(344,372)(345,371)(346,370)(347,369)(348,368)
(349,367)(350,366)(351,365)(352,364)(353,363)(354,362)(355,361)(356,360)
(357,359);
poly := sub<Sym(372)|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*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope