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Polytope of Type {42,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,16}*1344
Also Known As : {42,16|2}. if this polytope has another name.
Group : SmallGroup(1344,5836)
Rank : 3
Schlafli Type : {42,16}
Number of vertices, edges, etc : 42, 336, 16
Order of s0s1s2 : 336
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 : {42,8}*672
   3-fold quotients : {14,16}*448
   4-fold quotients : {42,4}*336a
   6-fold quotients : {14,8}*224
   7-fold quotients : {6,16}*192
   8-fold quotients : {42,2}*168
   12-fold quotients : {14,4}*112
   14-fold quotients : {6,8}*96
   16-fold quotients : {21,2}*84
   21-fold quotients : {2,16}*64
   24-fold quotients : {14,2}*56
   28-fold quotients : {6,4}*48a
   42-fold quotients : {2,8}*32
   48-fold quotients : {7,2}*28
   56-fold quotients : {6,2}*24
   84-fold quotients : {2,4}*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)( 44, 49)( 45, 48)( 46, 47)( 50, 57)
( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)( 66, 69)
( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)
( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)( 96,102)
( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)(115,125)
(116,124)(117,123)(118,122)(119,121)(128,133)(129,132)(130,131)(134,141)
(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)(150,153)
(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)
(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)(180,186)
(181,185)(182,184)(191,196)(192,195)(193,194)(197,204)(198,210)(199,209)
(200,208)(201,207)(202,206)(203,205)(212,217)(213,216)(214,215)(218,225)
(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)(233,238)(234,237)
(235,236)(239,246)(240,252)(241,251)(242,250)(243,249)(244,248)(245,247)
(254,259)(255,258)(256,257)(260,267)(261,273)(262,272)(263,271)(264,270)
(265,269)(266,268)(275,280)(276,279)(277,278)(281,288)(282,294)(283,293)
(284,292)(285,291)(286,290)(287,289)(296,301)(297,300)(298,299)(302,309)
(303,315)(304,314)(305,313)(306,312)(307,311)(308,310)(317,322)(318,321)
(319,320)(323,330)(324,336)(325,335)(326,334)(327,333)(328,332)(329,331);;
s1 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 16)
( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)( 27, 32)
( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 72)( 44, 71)( 45, 77)( 46, 76)
( 47, 75)( 48, 74)( 49, 73)( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)
( 55, 67)( 56, 66)( 57, 79)( 58, 78)( 59, 84)( 60, 83)( 61, 82)( 62, 81)
( 63, 80)( 85,135)( 86,134)( 87,140)( 88,139)( 89,138)( 90,137)( 91,136)
( 92,128)( 93,127)( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)( 99,142)
(100,141)(101,147)(102,146)(103,145)(104,144)(105,143)(106,156)(107,155)
(108,161)(109,160)(110,159)(111,158)(112,157)(113,149)(114,148)(115,154)
(116,153)(117,152)(118,151)(119,150)(120,163)(121,162)(122,168)(123,167)
(124,166)(125,165)(126,164)(169,261)(170,260)(171,266)(172,265)(173,264)
(174,263)(175,262)(176,254)(177,253)(178,259)(179,258)(180,257)(181,256)
(182,255)(183,268)(184,267)(185,273)(186,272)(187,271)(188,270)(189,269)
(190,282)(191,281)(192,287)(193,286)(194,285)(195,284)(196,283)(197,275)
(198,274)(199,280)(200,279)(201,278)(202,277)(203,276)(204,289)(205,288)
(206,294)(207,293)(208,292)(209,291)(210,290)(211,324)(212,323)(213,329)
(214,328)(215,327)(216,326)(217,325)(218,317)(219,316)(220,322)(221,321)
(222,320)(223,319)(224,318)(225,331)(226,330)(227,336)(228,335)(229,334)
(230,333)(231,332)(232,303)(233,302)(234,308)(235,307)(236,306)(237,305)
(238,304)(239,296)(240,295)(241,301)(242,300)(243,299)(244,298)(245,297)
(246,310)(247,309)(248,315)(249,314)(250,313)(251,312)(252,311);;
s2 := (  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)(  8,176)
(  9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)( 16,184)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)( 32,200)
( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)
( 41,209)( 42,210)( 43,232)( 44,233)( 45,234)( 46,235)( 47,236)( 48,237)
( 49,238)( 50,239)( 51,240)( 52,241)( 53,242)( 54,243)( 55,244)( 56,245)
( 57,246)( 58,247)( 59,248)( 60,249)( 61,250)( 62,251)( 63,252)( 64,211)
( 65,212)( 66,213)( 67,214)( 68,215)( 69,216)( 70,217)( 71,218)( 72,219)
( 73,220)( 74,221)( 75,222)( 76,223)( 77,224)( 78,225)( 79,226)( 80,227)
( 81,228)( 82,229)( 83,230)( 84,231)( 85,295)( 86,296)( 87,297)( 88,298)
( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)( 96,306)
( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)(104,314)
(105,315)(106,316)(107,317)(108,318)(109,319)(110,320)(111,321)(112,322)
(113,323)(114,324)(115,325)(116,326)(117,327)(118,328)(119,329)(120,330)
(121,331)(122,332)(123,333)(124,334)(125,335)(126,336)(127,253)(128,254)
(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)(136,262)
(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)(144,270)
(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)(152,278)
(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)
(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(336)!(  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)( 44, 49)( 45, 48)( 46, 47)
( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)
( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)
( 96,102)( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)
(115,125)(116,124)(117,123)(118,122)(119,121)(128,133)(129,132)(130,131)
(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)
(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)
(161,163)(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)
(180,186)(181,185)(182,184)(191,196)(192,195)(193,194)(197,204)(198,210)
(199,209)(200,208)(201,207)(202,206)(203,205)(212,217)(213,216)(214,215)
(218,225)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)(233,238)
(234,237)(235,236)(239,246)(240,252)(241,251)(242,250)(243,249)(244,248)
(245,247)(254,259)(255,258)(256,257)(260,267)(261,273)(262,272)(263,271)
(264,270)(265,269)(266,268)(275,280)(276,279)(277,278)(281,288)(282,294)
(283,293)(284,292)(285,291)(286,290)(287,289)(296,301)(297,300)(298,299)
(302,309)(303,315)(304,314)(305,313)(306,312)(307,311)(308,310)(317,322)
(318,321)(319,320)(323,330)(324,336)(325,335)(326,334)(327,333)(328,332)
(329,331);
s1 := Sym(336)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 16)( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)
( 27, 32)( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 72)( 44, 71)( 45, 77)
( 46, 76)( 47, 75)( 48, 74)( 49, 73)( 50, 65)( 51, 64)( 52, 70)( 53, 69)
( 54, 68)( 55, 67)( 56, 66)( 57, 79)( 58, 78)( 59, 84)( 60, 83)( 61, 82)
( 62, 81)( 63, 80)( 85,135)( 86,134)( 87,140)( 88,139)( 89,138)( 90,137)
( 91,136)( 92,128)( 93,127)( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)
( 99,142)(100,141)(101,147)(102,146)(103,145)(104,144)(105,143)(106,156)
(107,155)(108,161)(109,160)(110,159)(111,158)(112,157)(113,149)(114,148)
(115,154)(116,153)(117,152)(118,151)(119,150)(120,163)(121,162)(122,168)
(123,167)(124,166)(125,165)(126,164)(169,261)(170,260)(171,266)(172,265)
(173,264)(174,263)(175,262)(176,254)(177,253)(178,259)(179,258)(180,257)
(181,256)(182,255)(183,268)(184,267)(185,273)(186,272)(187,271)(188,270)
(189,269)(190,282)(191,281)(192,287)(193,286)(194,285)(195,284)(196,283)
(197,275)(198,274)(199,280)(200,279)(201,278)(202,277)(203,276)(204,289)
(205,288)(206,294)(207,293)(208,292)(209,291)(210,290)(211,324)(212,323)
(213,329)(214,328)(215,327)(216,326)(217,325)(218,317)(219,316)(220,322)
(221,321)(222,320)(223,319)(224,318)(225,331)(226,330)(227,336)(228,335)
(229,334)(230,333)(231,332)(232,303)(233,302)(234,308)(235,307)(236,306)
(237,305)(238,304)(239,296)(240,295)(241,301)(242,300)(243,299)(244,298)
(245,297)(246,310)(247,309)(248,315)(249,314)(250,313)(251,312)(252,311);
s2 := Sym(336)!(  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)
(  8,176)(  9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)
( 16,184)( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)
( 24,192)( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)
( 32,200)( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)
( 40,208)( 41,209)( 42,210)( 43,232)( 44,233)( 45,234)( 46,235)( 47,236)
( 48,237)( 49,238)( 50,239)( 51,240)( 52,241)( 53,242)( 54,243)( 55,244)
( 56,245)( 57,246)( 58,247)( 59,248)( 60,249)( 61,250)( 62,251)( 63,252)
( 64,211)( 65,212)( 66,213)( 67,214)( 68,215)( 69,216)( 70,217)( 71,218)
( 72,219)( 73,220)( 74,221)( 75,222)( 76,223)( 77,224)( 78,225)( 79,226)
( 80,227)( 81,228)( 82,229)( 83,230)( 84,231)( 85,295)( 86,296)( 87,297)
( 88,298)( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)
( 96,306)( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)
(104,314)(105,315)(106,316)(107,317)(108,318)(109,319)(110,320)(111,321)
(112,322)(113,323)(114,324)(115,325)(116,326)(117,327)(118,328)(119,329)
(120,330)(121,331)(122,332)(123,333)(124,334)(125,335)(126,336)(127,253)
(128,254)(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)
(136,262)(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)
(144,270)(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)
(152,278)(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)
(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)
(168,294);
poly := sub<Sym(336)|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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 >; 
 
References : None.
to this polytope