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Polytope of Type {21,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,8}*672
if this polytope has a name.
Group : SmallGroup(672,1086)
Rank : 3
Schlafli Type : {21,8}
Number of vertices, edges, etc : 42, 168, 16
Order of s0s1s2 : 84
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {21,8,2} of size 1344
Vertex Figure Of :
   {2,21,8} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {21,4}*336
   4-fold quotients : {21,4}*168
   7-fold quotients : {3,8}*96
   8-fold quotients : {21,2}*84
   14-fold quotients : {3,4}*48
   24-fold quotients : {7,2}*28
   28-fold quotients : {3,4}*24
   56-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {42,8}*1344b
Permutation Representation (GAP) :
s0 := (  3,  6)(  4,  5)(  7,  8)(  9, 49)( 10, 50)( 11, 54)( 12, 53)( 13, 52)
( 14, 51)( 15, 56)( 16, 55)( 17, 41)( 18, 42)( 19, 46)( 20, 45)( 21, 44)
( 22, 43)( 23, 48)( 24, 47)( 25, 33)( 26, 34)( 27, 38)( 28, 37)( 29, 36)
( 30, 35)( 31, 40)( 32, 39)( 57,113)( 58,114)( 59,118)( 60,117)( 61,116)
( 62,115)( 63,120)( 64,119)( 65,161)( 66,162)( 67,166)( 68,165)( 69,164)
( 70,163)( 71,168)( 72,167)( 73,153)( 74,154)( 75,158)( 76,157)( 77,156)
( 78,155)( 79,160)( 80,159)( 81,145)( 82,146)( 83,150)( 84,149)( 85,148)
( 86,147)( 87,152)( 88,151)( 89,137)( 90,138)( 91,142)( 92,141)( 93,140)
( 94,139)( 95,144)( 96,143)( 97,129)( 98,130)( 99,134)(100,133)(101,132)
(102,131)(103,136)(104,135)(105,121)(106,122)(107,126)(108,125)(109,124)
(110,123)(111,128)(112,127)(169,170)(171,173)(172,174)(177,218)(178,217)
(179,221)(180,222)(181,219)(182,220)(183,223)(184,224)(185,210)(186,209)
(187,213)(188,214)(189,211)(190,212)(191,215)(192,216)(193,202)(194,201)
(195,205)(196,206)(197,203)(198,204)(199,207)(200,208)(225,282)(226,281)
(227,285)(228,286)(229,283)(230,284)(231,287)(232,288)(233,330)(234,329)
(235,333)(236,334)(237,331)(238,332)(239,335)(240,336)(241,322)(242,321)
(243,325)(244,326)(245,323)(246,324)(247,327)(248,328)(249,314)(250,313)
(251,317)(252,318)(253,315)(254,316)(255,319)(256,320)(257,306)(258,305)
(259,309)(260,310)(261,307)(262,308)(263,311)(264,312)(265,298)(266,297)
(267,301)(268,302)(269,299)(270,300)(271,303)(272,304)(273,290)(274,289)
(275,293)(276,294)(277,291)(278,292)(279,295)(280,296);;
s1 := (  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 71)(  6, 72)(  7, 69)(  8, 70)
(  9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 63)( 14, 64)( 15, 61)( 16, 62)
( 17,105)( 18,106)( 19,108)( 20,107)( 21,111)( 22,112)( 23,109)( 24,110)
( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,103)( 30,104)( 31,101)( 32,102)
( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 95)( 38, 96)( 39, 93)( 40, 94)
( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 87)( 46, 88)( 47, 85)( 48, 86)
( 49, 73)( 50, 74)( 51, 76)( 52, 75)( 53, 79)( 54, 80)( 55, 77)( 56, 78)
(113,121)(114,122)(115,124)(116,123)(117,127)(118,128)(119,125)(120,126)
(129,161)(130,162)(131,164)(132,163)(133,167)(134,168)(135,165)(136,166)
(137,153)(138,154)(139,156)(140,155)(141,159)(142,160)(143,157)(144,158)
(147,148)(149,151)(150,152)(169,234)(170,233)(171,235)(172,236)(173,240)
(174,239)(175,238)(176,237)(177,226)(178,225)(179,227)(180,228)(181,232)
(182,231)(183,230)(184,229)(185,274)(186,273)(187,275)(188,276)(189,280)
(190,279)(191,278)(192,277)(193,266)(194,265)(195,267)(196,268)(197,272)
(198,271)(199,270)(200,269)(201,258)(202,257)(203,259)(204,260)(205,264)
(206,263)(207,262)(208,261)(209,250)(210,249)(211,251)(212,252)(213,256)
(214,255)(215,254)(216,253)(217,242)(218,241)(219,243)(220,244)(221,248)
(222,247)(223,246)(224,245)(281,290)(282,289)(283,291)(284,292)(285,296)
(286,295)(287,294)(288,293)(297,330)(298,329)(299,331)(300,332)(301,336)
(302,335)(303,334)(304,333)(305,322)(306,321)(307,323)(308,324)(309,328)
(310,327)(311,326)(312,325)(313,314)(317,320)(318,319);;
s2 := (  1,175)(  2,176)(  3,173)(  4,174)(  5,172)(  6,171)(  7,170)(  8,169)
(  9,183)( 10,184)( 11,181)( 12,182)( 13,180)( 14,179)( 15,178)( 16,177)
( 17,191)( 18,192)( 19,189)( 20,190)( 21,188)( 22,187)( 23,186)( 24,185)
( 25,199)( 26,200)( 27,197)( 28,198)( 29,196)( 30,195)( 31,194)( 32,193)
( 33,207)( 34,208)( 35,205)( 36,206)( 37,204)( 38,203)( 39,202)( 40,201)
( 41,215)( 42,216)( 43,213)( 44,214)( 45,212)( 46,211)( 47,210)( 48,209)
( 49,223)( 50,224)( 51,221)( 52,222)( 53,220)( 54,219)( 55,218)( 56,217)
( 57,231)( 58,232)( 59,229)( 60,230)( 61,228)( 62,227)( 63,226)( 64,225)
( 65,239)( 66,240)( 67,237)( 68,238)( 69,236)( 70,235)( 71,234)( 72,233)
( 73,247)( 74,248)( 75,245)( 76,246)( 77,244)( 78,243)( 79,242)( 80,241)
( 81,255)( 82,256)( 83,253)( 84,254)( 85,252)( 86,251)( 87,250)( 88,249)
( 89,263)( 90,264)( 91,261)( 92,262)( 93,260)( 94,259)( 95,258)( 96,257)
( 97,271)( 98,272)( 99,269)(100,270)(101,268)(102,267)(103,266)(104,265)
(105,279)(106,280)(107,277)(108,278)(109,276)(110,275)(111,274)(112,273)
(113,287)(114,288)(115,285)(116,286)(117,284)(118,283)(119,282)(120,281)
(121,295)(122,296)(123,293)(124,294)(125,292)(126,291)(127,290)(128,289)
(129,303)(130,304)(131,301)(132,302)(133,300)(134,299)(135,298)(136,297)
(137,311)(138,312)(139,309)(140,310)(141,308)(142,307)(143,306)(144,305)
(145,319)(146,320)(147,317)(148,318)(149,316)(150,315)(151,314)(152,313)
(153,327)(154,328)(155,325)(156,326)(157,324)(158,323)(159,322)(160,321)
(161,335)(162,336)(163,333)(164,334)(165,332)(166,331)(167,330)(168,329);;
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*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(336)!(  3,  6)(  4,  5)(  7,  8)(  9, 49)( 10, 50)( 11, 54)( 12, 53)
( 13, 52)( 14, 51)( 15, 56)( 16, 55)( 17, 41)( 18, 42)( 19, 46)( 20, 45)
( 21, 44)( 22, 43)( 23, 48)( 24, 47)( 25, 33)( 26, 34)( 27, 38)( 28, 37)
( 29, 36)( 30, 35)( 31, 40)( 32, 39)( 57,113)( 58,114)( 59,118)( 60,117)
( 61,116)( 62,115)( 63,120)( 64,119)( 65,161)( 66,162)( 67,166)( 68,165)
( 69,164)( 70,163)( 71,168)( 72,167)( 73,153)( 74,154)( 75,158)( 76,157)
( 77,156)( 78,155)( 79,160)( 80,159)( 81,145)( 82,146)( 83,150)( 84,149)
( 85,148)( 86,147)( 87,152)( 88,151)( 89,137)( 90,138)( 91,142)( 92,141)
( 93,140)( 94,139)( 95,144)( 96,143)( 97,129)( 98,130)( 99,134)(100,133)
(101,132)(102,131)(103,136)(104,135)(105,121)(106,122)(107,126)(108,125)
(109,124)(110,123)(111,128)(112,127)(169,170)(171,173)(172,174)(177,218)
(178,217)(179,221)(180,222)(181,219)(182,220)(183,223)(184,224)(185,210)
(186,209)(187,213)(188,214)(189,211)(190,212)(191,215)(192,216)(193,202)
(194,201)(195,205)(196,206)(197,203)(198,204)(199,207)(200,208)(225,282)
(226,281)(227,285)(228,286)(229,283)(230,284)(231,287)(232,288)(233,330)
(234,329)(235,333)(236,334)(237,331)(238,332)(239,335)(240,336)(241,322)
(242,321)(243,325)(244,326)(245,323)(246,324)(247,327)(248,328)(249,314)
(250,313)(251,317)(252,318)(253,315)(254,316)(255,319)(256,320)(257,306)
(258,305)(259,309)(260,310)(261,307)(262,308)(263,311)(264,312)(265,298)
(266,297)(267,301)(268,302)(269,299)(270,300)(271,303)(272,304)(273,290)
(274,289)(275,293)(276,294)(277,291)(278,292)(279,295)(280,296);
s1 := Sym(336)!(  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 71)(  6, 72)(  7, 69)
(  8, 70)(  9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 63)( 14, 64)( 15, 61)
( 16, 62)( 17,105)( 18,106)( 19,108)( 20,107)( 21,111)( 22,112)( 23,109)
( 24,110)( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,103)( 30,104)( 31,101)
( 32,102)( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 95)( 38, 96)( 39, 93)
( 40, 94)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 87)( 46, 88)( 47, 85)
( 48, 86)( 49, 73)( 50, 74)( 51, 76)( 52, 75)( 53, 79)( 54, 80)( 55, 77)
( 56, 78)(113,121)(114,122)(115,124)(116,123)(117,127)(118,128)(119,125)
(120,126)(129,161)(130,162)(131,164)(132,163)(133,167)(134,168)(135,165)
(136,166)(137,153)(138,154)(139,156)(140,155)(141,159)(142,160)(143,157)
(144,158)(147,148)(149,151)(150,152)(169,234)(170,233)(171,235)(172,236)
(173,240)(174,239)(175,238)(176,237)(177,226)(178,225)(179,227)(180,228)
(181,232)(182,231)(183,230)(184,229)(185,274)(186,273)(187,275)(188,276)
(189,280)(190,279)(191,278)(192,277)(193,266)(194,265)(195,267)(196,268)
(197,272)(198,271)(199,270)(200,269)(201,258)(202,257)(203,259)(204,260)
(205,264)(206,263)(207,262)(208,261)(209,250)(210,249)(211,251)(212,252)
(213,256)(214,255)(215,254)(216,253)(217,242)(218,241)(219,243)(220,244)
(221,248)(222,247)(223,246)(224,245)(281,290)(282,289)(283,291)(284,292)
(285,296)(286,295)(287,294)(288,293)(297,330)(298,329)(299,331)(300,332)
(301,336)(302,335)(303,334)(304,333)(305,322)(306,321)(307,323)(308,324)
(309,328)(310,327)(311,326)(312,325)(313,314)(317,320)(318,319);
s2 := Sym(336)!(  1,175)(  2,176)(  3,173)(  4,174)(  5,172)(  6,171)(  7,170)
(  8,169)(  9,183)( 10,184)( 11,181)( 12,182)( 13,180)( 14,179)( 15,178)
( 16,177)( 17,191)( 18,192)( 19,189)( 20,190)( 21,188)( 22,187)( 23,186)
( 24,185)( 25,199)( 26,200)( 27,197)( 28,198)( 29,196)( 30,195)( 31,194)
( 32,193)( 33,207)( 34,208)( 35,205)( 36,206)( 37,204)( 38,203)( 39,202)
( 40,201)( 41,215)( 42,216)( 43,213)( 44,214)( 45,212)( 46,211)( 47,210)
( 48,209)( 49,223)( 50,224)( 51,221)( 52,222)( 53,220)( 54,219)( 55,218)
( 56,217)( 57,231)( 58,232)( 59,229)( 60,230)( 61,228)( 62,227)( 63,226)
( 64,225)( 65,239)( 66,240)( 67,237)( 68,238)( 69,236)( 70,235)( 71,234)
( 72,233)( 73,247)( 74,248)( 75,245)( 76,246)( 77,244)( 78,243)( 79,242)
( 80,241)( 81,255)( 82,256)( 83,253)( 84,254)( 85,252)( 86,251)( 87,250)
( 88,249)( 89,263)( 90,264)( 91,261)( 92,262)( 93,260)( 94,259)( 95,258)
( 96,257)( 97,271)( 98,272)( 99,269)(100,270)(101,268)(102,267)(103,266)
(104,265)(105,279)(106,280)(107,277)(108,278)(109,276)(110,275)(111,274)
(112,273)(113,287)(114,288)(115,285)(116,286)(117,284)(118,283)(119,282)
(120,281)(121,295)(122,296)(123,293)(124,294)(125,292)(126,291)(127,290)
(128,289)(129,303)(130,304)(131,301)(132,302)(133,300)(134,299)(135,298)
(136,297)(137,311)(138,312)(139,309)(140,310)(141,308)(142,307)(143,306)
(144,305)(145,319)(146,320)(147,317)(148,318)(149,316)(150,315)(151,314)
(152,313)(153,327)(154,328)(155,325)(156,326)(157,324)(158,323)(159,322)
(160,321)(161,335)(162,336)(163,333)(164,334)(165,332)(166,331)(167,330)
(168,329);
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*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*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 >; 
 
References : None.
to this polytope