Polytope of Type {372}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {372}*744
Also Known As : 372-gon, {372}. if this polytope has another name.
Group : SmallGroup(744,36)
Rank : 2
Schlafli Type : {372}
Number of vertices, edges, etc : 372, 372
Order of s₀s₁ : 372
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {372,2} of size 1488
Vertex Figure Of :
   {2,372} of size 1488
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {186}*372
   3-fold quotients : {124}*248
   4-fold quotients : {93}*186
   6-fold quotients : {62}*124
   12-fold quotients : {31}*62
   31-fold quotients : {12}*24
   62-fold quotients : {6}*12
   93-fold quotients : {4}*8
   124-fold quotients : {3}*6
   186-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {744}*1488
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)(187,280)(188,310)(189,309)(190,308)
(191,307)(192,306)(193,305)(194,304)(195,303)(196,302)(197,301)(198,300)
(199,299)(200,298)(201,297)(202,296)(203,295)(204,294)(205,293)(206,292)
(207,291)(208,290)(209,289)(210,288)(211,287)(212,286)(213,285)(214,284)
(215,283)(216,282)(217,281)(218,342)(219,372)(220,371)(221,370)(222,369)
(223,368)(224,367)(225,366)(226,365)(227,364)(228,363)(229,362)(230,361)
(231,360)(232,359)(233,358)(234,357)(235,356)(236,355)(237,354)(238,353)
(239,352)(240,351)(241,350)(242,349)(243,348)(244,347)(245,346)(246,345)
(247,344)(248,343)(249,311)(250,341)(251,340)(252,339)(253,338)(254,337)
(255,336)(256,335)(257,334)(258,333)(259,332)(260,331)(261,330)(262,329)
(263,328)(264,327)(265,326)(266,325)(267,324)(268,323)(269,322)(270,321)
(271,320)(272,319)(273,318)(274,317)(275,316)(276,315)(277,314)(278,313)
(279,312);;
s1 := (  1,219)(  2,218)(  3,248)(  4,247)(  5,246)(  6,245)(  7,244)(  8,243)
(  9,242)( 10,241)( 11,240)( 12,239)( 13,238)( 14,237)( 15,236)( 16,235)
( 17,234)( 18,233)( 19,232)( 20,231)( 21,230)( 22,229)( 23,228)( 24,227)
( 25,226)( 26,225)( 27,224)( 28,223)( 29,222)( 30,221)( 31,220)( 32,188)
( 33,187)( 34,217)( 35,216)( 36,215)( 37,214)( 38,213)( 39,212)( 40,211)
( 41,210)( 42,209)( 43,208)( 44,207)( 45,206)( 46,205)( 47,204)( 48,203)
( 49,202)( 50,201)( 51,200)( 52,199)( 53,198)( 54,197)( 55,196)( 56,195)
( 57,194)( 58,193)( 59,192)( 60,191)( 61,190)( 62,189)( 63,250)( 64,249)
( 65,279)( 66,278)( 67,277)( 68,276)( 69,275)( 70,274)( 71,273)( 72,272)
( 73,271)( 74,270)( 75,269)( 76,268)( 77,267)( 78,266)( 79,265)( 80,264)
( 81,263)( 82,262)( 83,261)( 84,260)( 85,259)( 86,258)( 87,257)( 88,256)
( 89,255)( 90,254)( 91,253)( 92,252)( 93,251)( 94,312)( 95,311)( 96,341)
( 97,340)( 98,339)( 99,338)(100,337)(101,336)(102,335)(103,334)(104,333)
(105,332)(106,331)(107,330)(108,329)(109,328)(110,327)(111,326)(112,325)
(113,324)(114,323)(115,322)(116,321)(117,320)(118,319)(119,318)(120,317)
(121,316)(122,315)(123,314)(124,313)(125,281)(126,280)(127,310)(128,309)
(129,308)(130,307)(131,306)(132,305)(133,304)(134,303)(135,302)(136,301)
(137,300)(138,299)(139,298)(140,297)(141,296)(142,295)(143,294)(144,293)
(145,292)(146,291)(147,290)(148,289)(149,288)(150,287)(151,286)(152,285)
(153,284)(154,283)(155,282)(156,343)(157,342)(158,372)(159,371)(160,370)
(161,369)(162,368)(163,367)(164,366)(165,365)(166,364)(167,363)(168,362)
(169,361)(170,360)(171,359)(172,358)(173,357)(174,356)(175,355)(176,354)
(177,353)(178,352)(179,351)(180,350)(181,349)(182,348)(183,347)(184,346)
(185,345)(186,344);;
poly := Group([s0,s1]);;

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

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