Polytope of Type {342}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {342}*684
Also Known As : 342-gon, {342}. if this polytope has another name.
Group : SmallGroup(684,14)
Rank : 2
Schlafli Type : {342}
Number of vertices, edges, etc : 342, 342
Order of s0s1 : 342
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{342,2} of size 1368
Vertex Figure Of :
{2,342} of size 1368
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {171}*342
3-fold quotients : {114}*228
6-fold quotients : {57}*114
9-fold quotients : {38}*76
18-fold quotients : {19}*38
19-fold quotients : {18}*36
38-fold quotients : {9}*18
57-fold quotients : {6}*12
114-fold quotients : {3}*6
171-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {684}*1368
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 55)( 5, 57)( 6, 56)( 7, 52)( 8, 54)( 9, 53)( 10, 49)
( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 43)( 17, 45)( 18, 44)
( 19, 40)( 20, 42)( 21, 41)( 22, 37)( 23, 39)( 24, 38)( 25, 34)( 26, 36)
( 27, 35)( 28, 31)( 29, 33)( 30, 32)( 58,117)( 59,116)( 60,115)( 61,171)
( 62,170)( 63,169)( 64,168)( 65,167)( 66,166)( 67,165)( 68,164)( 69,163)
( 70,162)( 71,161)( 72,160)( 73,159)( 74,158)( 75,157)( 76,156)( 77,155)
( 78,154)( 79,153)( 80,152)( 81,151)( 82,150)( 83,149)( 84,148)( 85,147)
( 86,146)( 87,145)( 88,144)( 89,143)( 90,142)( 91,141)( 92,140)( 93,139)
( 94,138)( 95,137)( 96,136)( 97,135)( 98,134)( 99,133)(100,132)(101,131)
(102,130)(103,129)(104,128)(105,127)(106,126)(107,125)(108,124)(109,123)
(110,122)(111,121)(112,120)(113,119)(114,118)(173,174)(175,226)(176,228)
(177,227)(178,223)(179,225)(180,224)(181,220)(182,222)(183,221)(184,217)
(185,219)(186,218)(187,214)(188,216)(189,215)(190,211)(191,213)(192,212)
(193,208)(194,210)(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)
(201,203)(229,288)(230,287)(231,286)(232,342)(233,341)(234,340)(235,339)
(236,338)(237,337)(238,336)(239,335)(240,334)(241,333)(242,332)(243,331)
(244,330)(245,329)(246,328)(247,327)(248,326)(249,325)(250,324)(251,323)
(252,322)(253,321)(254,320)(255,319)(256,318)(257,317)(258,316)(259,315)
(260,314)(261,313)(262,312)(263,311)(264,310)(265,309)(266,308)(267,307)
(268,306)(269,305)(270,304)(271,303)(272,302)(273,301)(274,300)(275,299)
(276,298)(277,297)(278,296)(279,295)(280,294)(281,293)(282,292)(283,291)
(284,290)(285,289);;
s1 := ( 1,232)( 2,234)( 3,233)( 4,229)( 5,231)( 6,230)( 7,283)( 8,285)
( 9,284)( 10,280)( 11,282)( 12,281)( 13,277)( 14,279)( 15,278)( 16,274)
( 17,276)( 18,275)( 19,271)( 20,273)( 21,272)( 22,268)( 23,270)( 24,269)
( 25,265)( 26,267)( 27,266)( 28,262)( 29,264)( 30,263)( 31,259)( 32,261)
( 33,260)( 34,256)( 35,258)( 36,257)( 37,253)( 38,255)( 39,254)( 40,250)
( 41,252)( 42,251)( 43,247)( 44,249)( 45,248)( 46,244)( 47,246)( 48,245)
( 49,241)( 50,243)( 51,242)( 52,238)( 53,240)( 54,239)( 55,235)( 56,237)
( 57,236)( 58,175)( 59,177)( 60,176)( 61,172)( 62,174)( 63,173)( 64,226)
( 65,228)( 66,227)( 67,223)( 68,225)( 69,224)( 70,220)( 71,222)( 72,221)
( 73,217)( 74,219)( 75,218)( 76,214)( 77,216)( 78,215)( 79,211)( 80,213)
( 81,212)( 82,208)( 83,210)( 84,209)( 85,205)( 86,207)( 87,206)( 88,202)
( 89,204)( 90,203)( 91,199)( 92,201)( 93,200)( 94,196)( 95,198)( 96,197)
( 97,193)( 98,195)( 99,194)(100,190)(101,192)(102,191)(103,187)(104,189)
(105,188)(106,184)(107,186)(108,185)(109,181)(110,183)(111,182)(112,178)
(113,180)(114,179)(115,291)(116,290)(117,289)(118,288)(119,287)(120,286)
(121,342)(122,341)(123,340)(124,339)(125,338)(126,337)(127,336)(128,335)
(129,334)(130,333)(131,332)(132,331)(133,330)(134,329)(135,328)(136,327)
(137,326)(138,325)(139,324)(140,323)(141,322)(142,321)(143,320)(144,319)
(145,318)(146,317)(147,316)(148,315)(149,314)(150,313)(151,312)(152,311)
(153,310)(154,309)(155,308)(156,307)(157,306)(158,305)(159,304)(160,303)
(161,302)(162,301)(163,300)(164,299)(165,298)(166,297)(167,296)(168,295)
(169,294)(170,293)(171,292);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(342)!( 2, 3)( 4, 55)( 5, 57)( 6, 56)( 7, 52)( 8, 54)( 9, 53)
( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 43)( 17, 45)
( 18, 44)( 19, 40)( 20, 42)( 21, 41)( 22, 37)( 23, 39)( 24, 38)( 25, 34)
( 26, 36)( 27, 35)( 28, 31)( 29, 33)( 30, 32)( 58,117)( 59,116)( 60,115)
( 61,171)( 62,170)( 63,169)( 64,168)( 65,167)( 66,166)( 67,165)( 68,164)
( 69,163)( 70,162)( 71,161)( 72,160)( 73,159)( 74,158)( 75,157)( 76,156)
( 77,155)( 78,154)( 79,153)( 80,152)( 81,151)( 82,150)( 83,149)( 84,148)
( 85,147)( 86,146)( 87,145)( 88,144)( 89,143)( 90,142)( 91,141)( 92,140)
( 93,139)( 94,138)( 95,137)( 96,136)( 97,135)( 98,134)( 99,133)(100,132)
(101,131)(102,130)(103,129)(104,128)(105,127)(106,126)(107,125)(108,124)
(109,123)(110,122)(111,121)(112,120)(113,119)(114,118)(173,174)(175,226)
(176,228)(177,227)(178,223)(179,225)(180,224)(181,220)(182,222)(183,221)
(184,217)(185,219)(186,218)(187,214)(188,216)(189,215)(190,211)(191,213)
(192,212)(193,208)(194,210)(195,209)(196,205)(197,207)(198,206)(199,202)
(200,204)(201,203)(229,288)(230,287)(231,286)(232,342)(233,341)(234,340)
(235,339)(236,338)(237,337)(238,336)(239,335)(240,334)(241,333)(242,332)
(243,331)(244,330)(245,329)(246,328)(247,327)(248,326)(249,325)(250,324)
(251,323)(252,322)(253,321)(254,320)(255,319)(256,318)(257,317)(258,316)
(259,315)(260,314)(261,313)(262,312)(263,311)(264,310)(265,309)(266,308)
(267,307)(268,306)(269,305)(270,304)(271,303)(272,302)(273,301)(274,300)
(275,299)(276,298)(277,297)(278,296)(279,295)(280,294)(281,293)(282,292)
(283,291)(284,290)(285,289);
s1 := Sym(342)!( 1,232)( 2,234)( 3,233)( 4,229)( 5,231)( 6,230)( 7,283)
( 8,285)( 9,284)( 10,280)( 11,282)( 12,281)( 13,277)( 14,279)( 15,278)
( 16,274)( 17,276)( 18,275)( 19,271)( 20,273)( 21,272)( 22,268)( 23,270)
( 24,269)( 25,265)( 26,267)( 27,266)( 28,262)( 29,264)( 30,263)( 31,259)
( 32,261)( 33,260)( 34,256)( 35,258)( 36,257)( 37,253)( 38,255)( 39,254)
( 40,250)( 41,252)( 42,251)( 43,247)( 44,249)( 45,248)( 46,244)( 47,246)
( 48,245)( 49,241)( 50,243)( 51,242)( 52,238)( 53,240)( 54,239)( 55,235)
( 56,237)( 57,236)( 58,175)( 59,177)( 60,176)( 61,172)( 62,174)( 63,173)
( 64,226)( 65,228)( 66,227)( 67,223)( 68,225)( 69,224)( 70,220)( 71,222)
( 72,221)( 73,217)( 74,219)( 75,218)( 76,214)( 77,216)( 78,215)( 79,211)
( 80,213)( 81,212)( 82,208)( 83,210)( 84,209)( 85,205)( 86,207)( 87,206)
( 88,202)( 89,204)( 90,203)( 91,199)( 92,201)( 93,200)( 94,196)( 95,198)
( 96,197)( 97,193)( 98,195)( 99,194)(100,190)(101,192)(102,191)(103,187)
(104,189)(105,188)(106,184)(107,186)(108,185)(109,181)(110,183)(111,182)
(112,178)(113,180)(114,179)(115,291)(116,290)(117,289)(118,288)(119,287)
(120,286)(121,342)(122,341)(123,340)(124,339)(125,338)(126,337)(127,336)
(128,335)(129,334)(130,333)(131,332)(132,331)(133,330)(134,329)(135,328)
(136,327)(137,326)(138,325)(139,324)(140,323)(141,322)(142,321)(143,320)
(144,319)(145,318)(146,317)(147,316)(148,315)(149,314)(150,313)(151,312)
(152,311)(153,310)(154,309)(155,308)(156,307)(157,306)(158,305)(159,304)
(160,303)(161,302)(162,301)(163,300)(164,299)(165,298)(166,297)(167,296)
(168,295)(169,294)(170,293)(171,292);
poly := sub<Sym(342)|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 >;
References : None.
to this polytope