Polytope of Type {336,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {336,2}*1344
if this polytope has a name.
Group : SmallGroup(1344,5829)
Rank : 3
Schlafli Type : {336,2}
Number of vertices, edges, etc : 336, 336, 2
Order of s₀s₁s₂ : 336
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {168,2}*672
   3-fold quotients : {112,2}*448
   4-fold quotients : {84,2}*336
   6-fold quotients : {56,2}*224
   7-fold quotients : {48,2}*192
   8-fold quotients : {42,2}*168
   12-fold quotients : {28,2}*112
   14-fold quotients : {24,2}*96
   16-fold quotients : {21,2}*84
   21-fold quotients : {16,2}*64
   24-fold quotients : {14,2}*56
   28-fold quotients : {12,2}*48
   42-fold quotients : {8,2}*32
   48-fold quotients : {7,2}*28
   56-fold quotients : {6,2}*24
   84-fold quotients : {4,2}*16
   112-fold quotients : {3,2}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope