Polytope of Type {2,372}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,372}*1488
if this polytope has a name.
Group : SmallGroup(1488,195)
Rank : 3
Schlafli Type : {2,372}
Number of vertices, edges, etc : 2, 372, 372
Order of s₀s₁s₂ : 372
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {2,124}*496
   4-fold quotients : {2,93}*372
   6-fold quotients : {2,62}*248
   12-fold quotients : {2,31}*124
   31-fold quotients : {2,12}*48
   62-fold quotients : {2,6}*24
   93-fold quotients : {2,4}*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 := (1,2);;
s1 := (  4, 33)(  5, 32)(  6, 31)(  7, 30)(  8, 29)(  9, 28)( 10, 27)( 11, 26)
( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 34, 65)
( 35, 95)( 36, 94)( 37, 93)( 38, 92)( 39, 91)( 40, 90)( 41, 89)( 42, 88)
( 43, 87)( 44, 86)( 45, 85)( 46, 84)( 47, 83)( 48, 82)( 49, 81)( 50, 80)
( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 55, 75)( 56, 74)( 57, 73)( 58, 72)
( 59, 71)( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 64, 66)( 97,126)( 98,125)
( 99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)
(107,116)(108,115)(109,114)(110,113)(111,112)(127,158)(128,188)(129,187)
(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,180)(137,179)
(138,178)(139,177)(140,176)(141,175)(142,174)(143,173)(144,172)(145,171)
(146,170)(147,169)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)
(154,162)(155,161)(156,160)(157,159)(189,282)(190,312)(191,311)(192,310)
(193,309)(194,308)(195,307)(196,306)(197,305)(198,304)(199,303)(200,302)
(201,301)(202,300)(203,299)(204,298)(205,297)(206,296)(207,295)(208,294)
(209,293)(210,292)(211,291)(212,290)(213,289)(214,288)(215,287)(216,286)
(217,285)(218,284)(219,283)(220,344)(221,374)(222,373)(223,372)(224,371)
(225,370)(226,369)(227,368)(228,367)(229,366)(230,365)(231,364)(232,363)
(233,362)(234,361)(235,360)(236,359)(237,358)(238,357)(239,356)(240,355)
(241,354)(242,353)(243,352)(244,351)(245,350)(246,349)(247,348)(248,347)
(249,346)(250,345)(251,313)(252,343)(253,342)(254,341)(255,340)(256,339)
(257,338)(258,337)(259,336)(260,335)(261,334)(262,333)(263,332)(264,331)
(265,330)(266,329)(267,328)(268,327)(269,326)(270,325)(271,324)(272,323)
(273,322)(274,321)(275,320)(276,319)(277,318)(278,317)(279,316)(280,315)
(281,314);;
s2 := (  3,221)(  4,220)(  5,250)(  6,249)(  7,248)(  8,247)(  9,246)( 10,245)
( 11,244)( 12,243)( 13,242)( 14,241)( 15,240)( 16,239)( 17,238)( 18,237)
( 19,236)( 20,235)( 21,234)( 22,233)( 23,232)( 24,231)( 25,230)( 26,229)
( 27,228)( 28,227)( 29,226)( 30,225)( 31,224)( 32,223)( 33,222)( 34,190)
( 35,189)( 36,219)( 37,218)( 38,217)( 39,216)( 40,215)( 41,214)( 42,213)
( 43,212)( 44,211)( 45,210)( 46,209)( 47,208)( 48,207)( 49,206)( 50,205)
( 51,204)( 52,203)( 53,202)( 54,201)( 55,200)( 56,199)( 57,198)( 58,197)
( 59,196)( 60,195)( 61,194)( 62,193)( 63,192)( 64,191)( 65,252)( 66,251)
( 67,281)( 68,280)( 69,279)( 70,278)( 71,277)( 72,276)( 73,275)( 74,274)
( 75,273)( 76,272)( 77,271)( 78,270)( 79,269)( 80,268)( 81,267)( 82,266)
( 83,265)( 84,264)( 85,263)( 86,262)( 87,261)( 88,260)( 89,259)( 90,258)
( 91,257)( 92,256)( 93,255)( 94,254)( 95,253)( 96,314)( 97,313)( 98,343)
( 99,342)(100,341)(101,340)(102,339)(103,338)(104,337)(105,336)(106,335)
(107,334)(108,333)(109,332)(110,331)(111,330)(112,329)(113,328)(114,327)
(115,326)(116,325)(117,324)(118,323)(119,322)(120,321)(121,320)(122,319)
(123,318)(124,317)(125,316)(126,315)(127,283)(128,282)(129,312)(130,311)
(131,310)(132,309)(133,308)(134,307)(135,306)(136,305)(137,304)(138,303)
(139,302)(140,301)(141,300)(142,299)(143,298)(144,297)(145,296)(146,295)
(147,294)(148,293)(149,292)(150,291)(151,290)(152,289)(153,288)(154,287)
(155,286)(156,285)(157,284)(158,345)(159,344)(160,374)(161,373)(162,372)
(163,371)(164,370)(165,369)(166,368)(167,367)(168,366)(169,365)(170,364)
(171,363)(172,362)(173,361)(174,360)(175,359)(176,358)(177,357)(178,356)
(179,355)(180,354)(181,353)(182,352)(183,351)(184,350)(185,349)(186,348)
(187,347)(188,346);;
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*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(374)!(1,2);
s1 := Sym(374)!(  4, 33)(  5, 32)(  6, 31)(  7, 30)(  8, 29)(  9, 28)( 10, 27)
( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)
( 34, 65)( 35, 95)( 36, 94)( 37, 93)( 38, 92)( 39, 91)( 40, 90)( 41, 89)
( 42, 88)( 43, 87)( 44, 86)( 45, 85)( 46, 84)( 47, 83)( 48, 82)( 49, 81)
( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 55, 75)( 56, 74)( 57, 73)
( 58, 72)( 59, 71)( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 64, 66)( 97,126)
( 98,125)( 99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)
(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(127,158)(128,188)
(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,180)
(137,179)(138,178)(139,177)(140,176)(141,175)(142,174)(143,173)(144,172)
(145,171)(146,170)(147,169)(148,168)(149,167)(150,166)(151,165)(152,164)
(153,163)(154,162)(155,161)(156,160)(157,159)(189,282)(190,312)(191,311)
(192,310)(193,309)(194,308)(195,307)(196,306)(197,305)(198,304)(199,303)
(200,302)(201,301)(202,300)(203,299)(204,298)(205,297)(206,296)(207,295)
(208,294)(209,293)(210,292)(211,291)(212,290)(213,289)(214,288)(215,287)
(216,286)(217,285)(218,284)(219,283)(220,344)(221,374)(222,373)(223,372)
(224,371)(225,370)(226,369)(227,368)(228,367)(229,366)(230,365)(231,364)
(232,363)(233,362)(234,361)(235,360)(236,359)(237,358)(238,357)(239,356)
(240,355)(241,354)(242,353)(243,352)(244,351)(245,350)(246,349)(247,348)
(248,347)(249,346)(250,345)(251,313)(252,343)(253,342)(254,341)(255,340)
(256,339)(257,338)(258,337)(259,336)(260,335)(261,334)(262,333)(263,332)
(264,331)(265,330)(266,329)(267,328)(268,327)(269,326)(270,325)(271,324)
(272,323)(273,322)(274,321)(275,320)(276,319)(277,318)(278,317)(279,316)
(280,315)(281,314);
s2 := Sym(374)!(  3,221)(  4,220)(  5,250)(  6,249)(  7,248)(  8,247)(  9,246)
( 10,245)( 11,244)( 12,243)( 13,242)( 14,241)( 15,240)( 16,239)( 17,238)
( 18,237)( 19,236)( 20,235)( 21,234)( 22,233)( 23,232)( 24,231)( 25,230)
( 26,229)( 27,228)( 28,227)( 29,226)( 30,225)( 31,224)( 32,223)( 33,222)
( 34,190)( 35,189)( 36,219)( 37,218)( 38,217)( 39,216)( 40,215)( 41,214)
( 42,213)( 43,212)( 44,211)( 45,210)( 46,209)( 47,208)( 48,207)( 49,206)
( 50,205)( 51,204)( 52,203)( 53,202)( 54,201)( 55,200)( 56,199)( 57,198)
( 58,197)( 59,196)( 60,195)( 61,194)( 62,193)( 63,192)( 64,191)( 65,252)
( 66,251)( 67,281)( 68,280)( 69,279)( 70,278)( 71,277)( 72,276)( 73,275)
( 74,274)( 75,273)( 76,272)( 77,271)( 78,270)( 79,269)( 80,268)( 81,267)
( 82,266)( 83,265)( 84,264)( 85,263)( 86,262)( 87,261)( 88,260)( 89,259)
( 90,258)( 91,257)( 92,256)( 93,255)( 94,254)( 95,253)( 96,314)( 97,313)
( 98,343)( 99,342)(100,341)(101,340)(102,339)(103,338)(104,337)(105,336)
(106,335)(107,334)(108,333)(109,332)(110,331)(111,330)(112,329)(113,328)
(114,327)(115,326)(116,325)(117,324)(118,323)(119,322)(120,321)(121,320)
(122,319)(123,318)(124,317)(125,316)(126,315)(127,283)(128,282)(129,312)
(130,311)(131,310)(132,309)(133,308)(134,307)(135,306)(136,305)(137,304)
(138,303)(139,302)(140,301)(141,300)(142,299)(143,298)(144,297)(145,296)
(146,295)(147,294)(148,293)(149,292)(150,291)(151,290)(152,289)(153,288)
(154,287)(155,286)(156,285)(157,284)(158,345)(159,344)(160,374)(161,373)
(162,372)(163,371)(164,370)(165,369)(166,368)(167,367)(168,366)(169,365)
(170,364)(171,363)(172,362)(173,361)(174,360)(175,359)(176,358)(177,357)
(178,356)(179,355)(180,354)(181,353)(182,352)(183,351)(184,350)(185,349)
(186,348)(187,347)(188,346);
poly := sub<Sym(374)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope