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