Polytope of Type {8,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,42}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11412)
Rank : 3
Schlafli Type : {8,42}
Number of vertices, edges, etc : 16, 336, 84
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {4,42}*672
   4-fold quotients : {4,21}*336, {4,42}*336b, {4,42}*336c
   7-fold quotients : {8,6}*192c
   8-fold quotients : {4,21}*168, {2,42}*168
   14-fold quotients : {4,6}*96
   16-fold quotients : {2,21}*84
   24-fold quotients : {2,14}*56
   28-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {2,7}*28
   56-fold quotients : {4,3}*24, {2,6}*24
   112-fold quotients : {2,3}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope