Polytope of Type {174,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {174,4}*1392a
Also Known As : {174,4|2}. if this polytope has another name.
Group : SmallGroup(1392,173)
Rank : 3
Schlafli Type : {174,4}
Number of vertices, edges, etc : 174, 348, 4
Order of s₀s₁s₂ : 348
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 : {174,2}*696
   3-fold quotients : {58,4}*464
   4-fold quotients : {87,2}*348
   6-fold quotients : {58,2}*232
   12-fold quotients : {29,2}*116
   29-fold quotients : {6,4}*48a
   58-fold quotients : {6,2}*24
   87-fold quotients : {2,4}*16
   116-fold quotients : {3,2}*12
   174-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope