Polytope of Type {4,174}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,174}*1392a
Also Known As : {4,174|2}. if this polytope has another name.
Group : SmallGroup(1392,173)
Rank : 3
Schlafli Type : {4,174}
Number of vertices, edges, etc : 4, 348, 174
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 : {2,174}*696
3-fold quotients : {4,58}*464
4-fold quotients : {2,87}*348
6-fold quotients : {2,58}*232
12-fold quotients : {2,29}*116
29-fold quotients : {4,6}*48a
58-fold quotients : {2,6}*24
87-fold quotients : {4,2}*16
116-fold quotients : {2,3}*12
174-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (175,262)(176,263)(177,264)(178,265)(179,266)(180,267)(181,268)(182,269)
(183,270)(184,271)(185,272)(186,273)(187,274)(188,275)(189,276)(190,277)
(191,278)(192,279)(193,280)(194,281)(195,282)(196,283)(197,284)(198,285)
(199,286)(200,287)(201,288)(202,289)(203,290)(204,291)(205,292)(206,293)
(207,294)(208,295)(209,296)(210,297)(211,298)(212,299)(213,300)(214,301)
(215,302)(216,303)(217,304)(218,305)(219,306)(220,307)(221,308)(222,309)
(223,310)(224,311)(225,312)(226,313)(227,314)(228,315)(229,316)(230,317)
(231,318)(232,319)(233,320)(234,321)(235,322)(236,323)(237,324)(238,325)
(239,326)(240,327)(241,328)(242,329)(243,330)(244,331)(245,332)(246,333)
(247,334)(248,335)(249,336)(250,337)(251,338)(252,339)(253,340)(254,341)
(255,342)(256,343)(257,344)(258,345)(259,346)(260,347)(261,348);;
s1 := ( 1,175)( 2,203)( 3,202)( 4,201)( 5,200)( 6,199)( 7,198)( 8,197)
( 9,196)( 10,195)( 11,194)( 12,193)( 13,192)( 14,191)( 15,190)( 16,189)
( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,183)( 23,182)( 24,181)
( 25,180)( 26,179)( 27,178)( 28,177)( 29,176)( 30,233)( 31,261)( 32,260)
( 33,259)( 34,258)( 35,257)( 36,256)( 37,255)( 38,254)( 39,253)( 40,252)
( 41,251)( 42,250)( 43,249)( 44,248)( 45,247)( 46,246)( 47,245)( 48,244)
( 49,243)( 50,242)( 51,241)( 52,240)( 53,239)( 54,238)( 55,237)( 56,236)
( 57,235)( 58,234)( 59,204)( 60,232)( 61,231)( 62,230)( 63,229)( 64,228)
( 65,227)( 66,226)( 67,225)( 68,224)( 69,223)( 70,222)( 71,221)( 72,220)
( 73,219)( 74,218)( 75,217)( 76,216)( 77,215)( 78,214)( 79,213)( 80,212)
( 81,211)( 82,210)( 83,209)( 84,208)( 85,207)( 86,206)( 87,205)( 88,262)
( 89,290)( 90,289)( 91,288)( 92,287)( 93,286)( 94,285)( 95,284)( 96,283)
( 97,282)( 98,281)( 99,280)(100,279)(101,278)(102,277)(103,276)(104,275)
(105,274)(106,273)(107,272)(108,271)(109,270)(110,269)(111,268)(112,267)
(113,266)(114,265)(115,264)(116,263)(117,320)(118,348)(119,347)(120,346)
(121,345)(122,344)(123,343)(124,342)(125,341)(126,340)(127,339)(128,338)
(129,337)(130,336)(131,335)(132,334)(133,333)(134,332)(135,331)(136,330)
(137,329)(138,328)(139,327)(140,326)(141,325)(142,324)(143,323)(144,322)
(145,321)(146,291)(147,319)(148,318)(149,317)(150,316)(151,315)(152,314)
(153,313)(154,312)(155,311)(156,310)(157,309)(158,308)(159,307)(160,306)
(161,305)(162,304)(163,303)(164,302)(165,301)(166,300)(167,299)(168,298)
(169,297)(170,296)(171,295)(172,294)(173,293)(174,292);;
s2 := ( 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)(175,205)(176,204)
(177,232)(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)(184,225)
(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)
(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)(200,209)
(201,208)(202,207)(203,206)(233,234)(235,261)(236,260)(237,259)(238,258)
(239,257)(240,256)(241,255)(242,254)(243,253)(244,252)(245,251)(246,250)
(247,249)(262,292)(263,291)(264,319)(265,318)(266,317)(267,316)(268,315)
(269,314)(270,313)(271,312)(272,311)(273,310)(274,309)(275,308)(276,307)
(277,306)(278,305)(279,304)(280,303)(281,302)(282,301)(283,300)(284,299)
(285,298)(286,297)(287,296)(288,295)(289,294)(290,293)(320,321)(322,348)
(323,347)(324,346)(325,345)(326,344)(327,343)(328,342)(329,341)(330,340)
(331,339)(332,338)(333,337)(334,336);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(348)!(175,262)(176,263)(177,264)(178,265)(179,266)(180,267)(181,268)
(182,269)(183,270)(184,271)(185,272)(186,273)(187,274)(188,275)(189,276)
(190,277)(191,278)(192,279)(193,280)(194,281)(195,282)(196,283)(197,284)
(198,285)(199,286)(200,287)(201,288)(202,289)(203,290)(204,291)(205,292)
(206,293)(207,294)(208,295)(209,296)(210,297)(211,298)(212,299)(213,300)
(214,301)(215,302)(216,303)(217,304)(218,305)(219,306)(220,307)(221,308)
(222,309)(223,310)(224,311)(225,312)(226,313)(227,314)(228,315)(229,316)
(230,317)(231,318)(232,319)(233,320)(234,321)(235,322)(236,323)(237,324)
(238,325)(239,326)(240,327)(241,328)(242,329)(243,330)(244,331)(245,332)
(246,333)(247,334)(248,335)(249,336)(250,337)(251,338)(252,339)(253,340)
(254,341)(255,342)(256,343)(257,344)(258,345)(259,346)(260,347)(261,348);
s1 := Sym(348)!( 1,175)( 2,203)( 3,202)( 4,201)( 5,200)( 6,199)( 7,198)
( 8,197)( 9,196)( 10,195)( 11,194)( 12,193)( 13,192)( 14,191)( 15,190)
( 16,189)( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,183)( 23,182)
( 24,181)( 25,180)( 26,179)( 27,178)( 28,177)( 29,176)( 30,233)( 31,261)
( 32,260)( 33,259)( 34,258)( 35,257)( 36,256)( 37,255)( 38,254)( 39,253)
( 40,252)( 41,251)( 42,250)( 43,249)( 44,248)( 45,247)( 46,246)( 47,245)
( 48,244)( 49,243)( 50,242)( 51,241)( 52,240)( 53,239)( 54,238)( 55,237)
( 56,236)( 57,235)( 58,234)( 59,204)( 60,232)( 61,231)( 62,230)( 63,229)
( 64,228)( 65,227)( 66,226)( 67,225)( 68,224)( 69,223)( 70,222)( 71,221)
( 72,220)( 73,219)( 74,218)( 75,217)( 76,216)( 77,215)( 78,214)( 79,213)
( 80,212)( 81,211)( 82,210)( 83,209)( 84,208)( 85,207)( 86,206)( 87,205)
( 88,262)( 89,290)( 90,289)( 91,288)( 92,287)( 93,286)( 94,285)( 95,284)
( 96,283)( 97,282)( 98,281)( 99,280)(100,279)(101,278)(102,277)(103,276)
(104,275)(105,274)(106,273)(107,272)(108,271)(109,270)(110,269)(111,268)
(112,267)(113,266)(114,265)(115,264)(116,263)(117,320)(118,348)(119,347)
(120,346)(121,345)(122,344)(123,343)(124,342)(125,341)(126,340)(127,339)
(128,338)(129,337)(130,336)(131,335)(132,334)(133,333)(134,332)(135,331)
(136,330)(137,329)(138,328)(139,327)(140,326)(141,325)(142,324)(143,323)
(144,322)(145,321)(146,291)(147,319)(148,318)(149,317)(150,316)(151,315)
(152,314)(153,313)(154,312)(155,311)(156,310)(157,309)(158,308)(159,307)
(160,306)(161,305)(162,304)(163,303)(164,302)(165,301)(166,300)(167,299)
(168,298)(169,297)(170,296)(171,295)(172,294)(173,293)(174,292);
s2 := 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)(175,205)
(176,204)(177,232)(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)
(184,225)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)
(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)
(200,209)(201,208)(202,207)(203,206)(233,234)(235,261)(236,260)(237,259)
(238,258)(239,257)(240,256)(241,255)(242,254)(243,253)(244,252)(245,251)
(246,250)(247,249)(262,292)(263,291)(264,319)(265,318)(266,317)(267,316)
(268,315)(269,314)(270,313)(271,312)(272,311)(273,310)(274,309)(275,308)
(276,307)(277,306)(278,305)(279,304)(280,303)(281,302)(282,301)(283,300)
(284,299)(285,298)(286,297)(287,296)(288,295)(289,294)(290,293)(320,321)
(322,348)(323,347)(324,346)(325,345)(326,344)(327,343)(328,342)(329,341)
(330,340)(331,339)(332,338)(333,337)(334,336);
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*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 >;
References : None.
to this polytope