Polytope of Type {42,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,8}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11412)
Rank : 3
Schlafli Type : {42,8}
Number of vertices, edges, etc : 84, 336, 16
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   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 : {42,4}*672
   4-fold quotients : {21,4}*336, {42,4}*336b, {42,4}*336c
   7-fold quotients : {6,8}*192c
   8-fold quotients : {21,4}*168, {42,2}*168
   14-fold quotients : {6,4}*96
   16-fold quotients : {21,2}*84
   24-fold quotients : {14,2}*56
   28-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {7,2}*28
   56-fold quotients : {3,4}*24, {6,2}*24
   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 := (  3,  4)(  5,  7)(  6,  8)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 55)
( 14, 56)( 15, 53)( 16, 54)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 47)
( 22, 48)( 23, 45)( 24, 46)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 29, 39)
( 30, 40)( 31, 37)( 32, 38)( 57,113)( 58,114)( 59,116)( 60,115)( 61,119)
( 62,120)( 63,117)( 64,118)( 65,161)( 66,162)( 67,164)( 68,163)( 69,167)
( 70,168)( 71,165)( 72,166)( 73,153)( 74,154)( 75,156)( 76,155)( 77,159)
( 78,160)( 79,157)( 80,158)( 81,145)( 82,146)( 83,148)( 84,147)( 85,151)
( 86,152)( 87,149)( 88,150)( 89,137)( 90,138)( 91,140)( 92,139)( 93,143)
( 94,144)( 95,141)( 96,142)( 97,129)( 98,130)( 99,132)(100,131)(101,135)
(102,136)(103,133)(104,134)(105,121)(106,122)(107,124)(108,123)(109,127)
(110,128)(111,125)(112,126)(169,170)(173,176)(174,175)(177,218)(178,217)
(179,219)(180,220)(181,224)(182,223)(183,222)(184,221)(185,210)(186,209)
(187,211)(188,212)(189,216)(190,215)(191,214)(192,213)(193,202)(194,201)
(195,203)(196,204)(197,208)(198,207)(199,206)(200,205)(225,282)(226,281)
(227,283)(228,284)(229,288)(230,287)(231,286)(232,285)(233,330)(234,329)
(235,331)(236,332)(237,336)(238,335)(239,334)(240,333)(241,322)(242,321)
(243,323)(244,324)(245,328)(246,327)(247,326)(248,325)(249,314)(250,313)
(251,315)(252,316)(253,320)(254,319)(255,318)(256,317)(257,306)(258,305)
(259,307)(260,308)(261,312)(262,311)(263,310)(264,309)(265,298)(266,297)
(267,299)(268,300)(269,304)(270,303)(271,302)(272,301)(273,290)(274,289)
(275,291)(276,292)(277,296)(278,295)(279,294)(280,293);;
s1 := (  1, 65)(  2, 66)(  3, 71)(  4, 72)(  5, 70)(  6, 69)(  7, 67)(  8, 68)
(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)( 16, 60)
( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)( 24,108)
( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)( 32,100)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)( 48, 84)
( 49, 73)( 50, 74)( 51, 79)( 52, 80)( 53, 78)( 54, 77)( 55, 75)( 56, 76)
(113,121)(114,122)(115,127)(116,128)(117,126)(118,125)(119,123)(120,124)
(129,161)(130,162)(131,167)(132,168)(133,166)(134,165)(135,163)(136,164)
(137,153)(138,154)(139,159)(140,160)(141,158)(142,157)(143,155)(144,156)
(147,151)(148,152)(149,150)(169,233)(170,234)(171,239)(172,240)(173,238)
(174,237)(175,235)(176,236)(177,225)(178,226)(179,231)(180,232)(181,230)
(182,229)(183,227)(184,228)(185,273)(186,274)(187,279)(188,280)(189,278)
(190,277)(191,275)(192,276)(193,265)(194,266)(195,271)(196,272)(197,270)
(198,269)(199,267)(200,268)(201,257)(202,258)(203,263)(204,264)(205,262)
(206,261)(207,259)(208,260)(209,249)(210,250)(211,255)(212,256)(213,254)
(214,253)(215,251)(216,252)(217,241)(218,242)(219,247)(220,248)(221,246)
(222,245)(223,243)(224,244)(281,289)(282,290)(283,295)(284,296)(285,294)
(286,293)(287,291)(288,292)(297,329)(298,330)(299,335)(300,336)(301,334)
(302,333)(303,331)(304,332)(305,321)(306,322)(307,327)(308,328)(309,326)
(310,325)(311,323)(312,324)(315,319)(316,320)(317,318);;
s2 := (  1,171)(  2,172)(  3,170)(  4,169)(  5,175)(  6,176)(  7,174)(  8,173)
(  9,179)( 10,180)( 11,178)( 12,177)( 13,183)( 14,184)( 15,182)( 16,181)
( 17,187)( 18,188)( 19,186)( 20,185)( 21,191)( 22,192)( 23,190)( 24,189)
( 25,195)( 26,196)( 27,194)( 28,193)( 29,199)( 30,200)( 31,198)( 32,197)
( 33,203)( 34,204)( 35,202)( 36,201)( 37,207)( 38,208)( 39,206)( 40,205)
( 41,211)( 42,212)( 43,210)( 44,209)( 45,215)( 46,216)( 47,214)( 48,213)
( 49,219)( 50,220)( 51,218)( 52,217)( 53,223)( 54,224)( 55,222)( 56,221)
( 57,227)( 58,228)( 59,226)( 60,225)( 61,231)( 62,232)( 63,230)( 64,229)
( 65,235)( 66,236)( 67,234)( 68,233)( 69,239)( 70,240)( 71,238)( 72,237)
( 73,243)( 74,244)( 75,242)( 76,241)( 77,247)( 78,248)( 79,246)( 80,245)
( 81,251)( 82,252)( 83,250)( 84,249)( 85,255)( 86,256)( 87,254)( 88,253)
( 89,259)( 90,260)( 91,258)( 92,257)( 93,263)( 94,264)( 95,262)( 96,261)
( 97,267)( 98,268)( 99,266)(100,265)(101,271)(102,272)(103,270)(104,269)
(105,275)(106,276)(107,274)(108,273)(109,279)(110,280)(111,278)(112,277)
(113,283)(114,284)(115,282)(116,281)(117,287)(118,288)(119,286)(120,285)
(121,291)(122,292)(123,290)(124,289)(125,295)(126,296)(127,294)(128,293)
(129,299)(130,300)(131,298)(132,297)(133,303)(134,304)(135,302)(136,301)
(137,307)(138,308)(139,306)(140,305)(141,311)(142,312)(143,310)(144,309)
(145,315)(146,316)(147,314)(148,313)(149,319)(150,320)(151,318)(152,317)
(153,323)(154,324)(155,322)(156,321)(157,327)(158,328)(159,326)(160,325)
(161,331)(162,332)(163,330)(164,329)(165,335)(166,336)(167,334)(168,333);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!(  3,  4)(  5,  7)(  6,  8)(  9, 49)( 10, 50)( 11, 52)( 12, 51)
( 13, 55)( 14, 56)( 15, 53)( 16, 54)( 17, 41)( 18, 42)( 19, 44)( 20, 43)
( 21, 47)( 22, 48)( 23, 45)( 24, 46)( 25, 33)( 26, 34)( 27, 36)( 28, 35)
( 29, 39)( 30, 40)( 31, 37)( 32, 38)( 57,113)( 58,114)( 59,116)( 60,115)
( 61,119)( 62,120)( 63,117)( 64,118)( 65,161)( 66,162)( 67,164)( 68,163)
( 69,167)( 70,168)( 71,165)( 72,166)( 73,153)( 74,154)( 75,156)( 76,155)
( 77,159)( 78,160)( 79,157)( 80,158)( 81,145)( 82,146)( 83,148)( 84,147)
( 85,151)( 86,152)( 87,149)( 88,150)( 89,137)( 90,138)( 91,140)( 92,139)
( 93,143)( 94,144)( 95,141)( 96,142)( 97,129)( 98,130)( 99,132)(100,131)
(101,135)(102,136)(103,133)(104,134)(105,121)(106,122)(107,124)(108,123)
(109,127)(110,128)(111,125)(112,126)(169,170)(173,176)(174,175)(177,218)
(178,217)(179,219)(180,220)(181,224)(182,223)(183,222)(184,221)(185,210)
(186,209)(187,211)(188,212)(189,216)(190,215)(191,214)(192,213)(193,202)
(194,201)(195,203)(196,204)(197,208)(198,207)(199,206)(200,205)(225,282)
(226,281)(227,283)(228,284)(229,288)(230,287)(231,286)(232,285)(233,330)
(234,329)(235,331)(236,332)(237,336)(238,335)(239,334)(240,333)(241,322)
(242,321)(243,323)(244,324)(245,328)(246,327)(247,326)(248,325)(249,314)
(250,313)(251,315)(252,316)(253,320)(254,319)(255,318)(256,317)(257,306)
(258,305)(259,307)(260,308)(261,312)(262,311)(263,310)(264,309)(265,298)
(266,297)(267,299)(268,300)(269,304)(270,303)(271,302)(272,301)(273,290)
(274,289)(275,291)(276,292)(277,296)(278,295)(279,294)(280,293);
s1 := Sym(336)!(  1, 65)(  2, 66)(  3, 71)(  4, 72)(  5, 70)(  6, 69)(  7, 67)
(  8, 68)(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)
( 16, 60)( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)
( 24,108)( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)
( 32,100)( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)
( 40, 92)( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)
( 48, 84)( 49, 73)( 50, 74)( 51, 79)( 52, 80)( 53, 78)( 54, 77)( 55, 75)
( 56, 76)(113,121)(114,122)(115,127)(116,128)(117,126)(118,125)(119,123)
(120,124)(129,161)(130,162)(131,167)(132,168)(133,166)(134,165)(135,163)
(136,164)(137,153)(138,154)(139,159)(140,160)(141,158)(142,157)(143,155)
(144,156)(147,151)(148,152)(149,150)(169,233)(170,234)(171,239)(172,240)
(173,238)(174,237)(175,235)(176,236)(177,225)(178,226)(179,231)(180,232)
(181,230)(182,229)(183,227)(184,228)(185,273)(186,274)(187,279)(188,280)
(189,278)(190,277)(191,275)(192,276)(193,265)(194,266)(195,271)(196,272)
(197,270)(198,269)(199,267)(200,268)(201,257)(202,258)(203,263)(204,264)
(205,262)(206,261)(207,259)(208,260)(209,249)(210,250)(211,255)(212,256)
(213,254)(214,253)(215,251)(216,252)(217,241)(218,242)(219,247)(220,248)
(221,246)(222,245)(223,243)(224,244)(281,289)(282,290)(283,295)(284,296)
(285,294)(286,293)(287,291)(288,292)(297,329)(298,330)(299,335)(300,336)
(301,334)(302,333)(303,331)(304,332)(305,321)(306,322)(307,327)(308,328)
(309,326)(310,325)(311,323)(312,324)(315,319)(316,320)(317,318);
s2 := Sym(336)!(  1,171)(  2,172)(  3,170)(  4,169)(  5,175)(  6,176)(  7,174)
(  8,173)(  9,179)( 10,180)( 11,178)( 12,177)( 13,183)( 14,184)( 15,182)
( 16,181)( 17,187)( 18,188)( 19,186)( 20,185)( 21,191)( 22,192)( 23,190)
( 24,189)( 25,195)( 26,196)( 27,194)( 28,193)( 29,199)( 30,200)( 31,198)
( 32,197)( 33,203)( 34,204)( 35,202)( 36,201)( 37,207)( 38,208)( 39,206)
( 40,205)( 41,211)( 42,212)( 43,210)( 44,209)( 45,215)( 46,216)( 47,214)
( 48,213)( 49,219)( 50,220)( 51,218)( 52,217)( 53,223)( 54,224)( 55,222)
( 56,221)( 57,227)( 58,228)( 59,226)( 60,225)( 61,231)( 62,232)( 63,230)
( 64,229)( 65,235)( 66,236)( 67,234)( 68,233)( 69,239)( 70,240)( 71,238)
( 72,237)( 73,243)( 74,244)( 75,242)( 76,241)( 77,247)( 78,248)( 79,246)
( 80,245)( 81,251)( 82,252)( 83,250)( 84,249)( 85,255)( 86,256)( 87,254)
( 88,253)( 89,259)( 90,260)( 91,258)( 92,257)( 93,263)( 94,264)( 95,262)
( 96,261)( 97,267)( 98,268)( 99,266)(100,265)(101,271)(102,272)(103,270)
(104,269)(105,275)(106,276)(107,274)(108,273)(109,279)(110,280)(111,278)
(112,277)(113,283)(114,284)(115,282)(116,281)(117,287)(118,288)(119,286)
(120,285)(121,291)(122,292)(123,290)(124,289)(125,295)(126,296)(127,294)
(128,293)(129,299)(130,300)(131,298)(132,297)(133,303)(134,304)(135,302)
(136,301)(137,307)(138,308)(139,306)(140,305)(141,311)(142,312)(143,310)
(144,309)(145,315)(146,316)(147,314)(148,313)(149,319)(150,320)(151,318)
(152,317)(153,323)(154,324)(155,322)(156,321)(157,327)(158,328)(159,326)
(160,325)(161,331)(162,332)(163,330)(164,329)(165,335)(166,336)(167,334)
(168,333);
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, s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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