Polytope of Type {2,21,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,21,8}*1344
if this polytope has a name.
Group : SmallGroup(1344,11408)
Rank : 4
Schlafli Type : {2,21,8}
Number of vertices, edges, etc : 2, 42, 168, 16
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {2,21,4}*672
   4-fold quotients : {2,21,4}*336
   7-fold quotients : {2,3,8}*192
   8-fold quotients : {2,21,2}*168
   14-fold quotients : {2,3,4}*96
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {2,3,4}*48
   56-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  5,  8)(  6,  7)(  9, 10)( 11, 51)( 12, 52)( 13, 56)( 14, 55)( 15, 54)
( 16, 53)( 17, 58)( 18, 57)( 19, 43)( 20, 44)( 21, 48)( 22, 47)( 23, 46)
( 24, 45)( 25, 50)( 26, 49)( 27, 35)( 28, 36)( 29, 40)( 30, 39)( 31, 38)
( 32, 37)( 33, 42)( 34, 41)( 59,115)( 60,116)( 61,120)( 62,119)( 63,118)
( 64,117)( 65,122)( 66,121)( 67,163)( 68,164)( 69,168)( 70,167)( 71,166)
( 72,165)( 73,170)( 74,169)( 75,155)( 76,156)( 77,160)( 78,159)( 79,158)
( 80,157)( 81,162)( 82,161)( 83,147)( 84,148)( 85,152)( 86,151)( 87,150)
( 88,149)( 89,154)( 90,153)( 91,139)( 92,140)( 93,144)( 94,143)( 95,142)
( 96,141)( 97,146)( 98,145)( 99,131)(100,132)(101,136)(102,135)(103,134)
(104,133)(105,138)(106,137)(107,123)(108,124)(109,128)(110,127)(111,126)
(112,125)(113,130)(114,129)(171,172)(173,175)(174,176)(179,220)(180,219)
(181,223)(182,224)(183,221)(184,222)(185,225)(186,226)(187,212)(188,211)
(189,215)(190,216)(191,213)(192,214)(193,217)(194,218)(195,204)(196,203)
(197,207)(198,208)(199,205)(200,206)(201,209)(202,210)(227,284)(228,283)
(229,287)(230,288)(231,285)(232,286)(233,289)(234,290)(235,332)(236,331)
(237,335)(238,336)(239,333)(240,334)(241,337)(242,338)(243,324)(244,323)
(245,327)(246,328)(247,325)(248,326)(249,329)(250,330)(251,316)(252,315)
(253,319)(254,320)(255,317)(256,318)(257,321)(258,322)(259,308)(260,307)
(261,311)(262,312)(263,309)(264,310)(265,313)(266,314)(267,300)(268,299)
(269,303)(270,304)(271,301)(272,302)(273,305)(274,306)(275,292)(276,291)
(277,295)(278,296)(279,293)(280,294)(281,297)(282,298);;
s2 := (  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 73)(  8, 74)(  9, 71)( 10, 72)
( 11, 59)( 12, 60)( 13, 62)( 14, 61)( 15, 65)( 16, 66)( 17, 63)( 18, 64)
( 19,107)( 20,108)( 21,110)( 22,109)( 23,113)( 24,114)( 25,111)( 26,112)
( 27, 99)( 28,100)( 29,102)( 30,101)( 31,105)( 32,106)( 33,103)( 34,104)
( 35, 91)( 36, 92)( 37, 94)( 38, 93)( 39, 97)( 40, 98)( 41, 95)( 42, 96)
( 43, 83)( 44, 84)( 45, 86)( 46, 85)( 47, 89)( 48, 90)( 49, 87)( 50, 88)
( 51, 75)( 52, 76)( 53, 78)( 54, 77)( 55, 81)( 56, 82)( 57, 79)( 58, 80)
(115,123)(116,124)(117,126)(118,125)(119,129)(120,130)(121,127)(122,128)
(131,163)(132,164)(133,166)(134,165)(135,169)(136,170)(137,167)(138,168)
(139,155)(140,156)(141,158)(142,157)(143,161)(144,162)(145,159)(146,160)
(149,150)(151,153)(152,154)(171,236)(172,235)(173,237)(174,238)(175,242)
(176,241)(177,240)(178,239)(179,228)(180,227)(181,229)(182,230)(183,234)
(184,233)(185,232)(186,231)(187,276)(188,275)(189,277)(190,278)(191,282)
(192,281)(193,280)(194,279)(195,268)(196,267)(197,269)(198,270)(199,274)
(200,273)(201,272)(202,271)(203,260)(204,259)(205,261)(206,262)(207,266)
(208,265)(209,264)(210,263)(211,252)(212,251)(213,253)(214,254)(215,258)
(216,257)(217,256)(218,255)(219,244)(220,243)(221,245)(222,246)(223,250)
(224,249)(225,248)(226,247)(283,292)(284,291)(285,293)(286,294)(287,298)
(288,297)(289,296)(290,295)(299,332)(300,331)(301,333)(302,334)(303,338)
(304,337)(305,336)(306,335)(307,324)(308,323)(309,325)(310,326)(311,330)
(312,329)(313,328)(314,327)(315,316)(319,322)(320,321);;
s3 := (  3,177)(  4,178)(  5,175)(  6,176)(  7,174)(  8,173)(  9,172)( 10,171)
( 11,185)( 12,186)( 13,183)( 14,184)( 15,182)( 16,181)( 17,180)( 18,179)
( 19,193)( 20,194)( 21,191)( 22,192)( 23,190)( 24,189)( 25,188)( 26,187)
( 27,201)( 28,202)( 29,199)( 30,200)( 31,198)( 32,197)( 33,196)( 34,195)
( 35,209)( 36,210)( 37,207)( 38,208)( 39,206)( 40,205)( 41,204)( 42,203)
( 43,217)( 44,218)( 45,215)( 46,216)( 47,214)( 48,213)( 49,212)( 50,211)
( 51,225)( 52,226)( 53,223)( 54,224)( 55,222)( 56,221)( 57,220)( 58,219)
( 59,233)( 60,234)( 61,231)( 62,232)( 63,230)( 64,229)( 65,228)( 66,227)
( 67,241)( 68,242)( 69,239)( 70,240)( 71,238)( 72,237)( 73,236)( 74,235)
( 75,249)( 76,250)( 77,247)( 78,248)( 79,246)( 80,245)( 81,244)( 82,243)
( 83,257)( 84,258)( 85,255)( 86,256)( 87,254)( 88,253)( 89,252)( 90,251)
( 91,265)( 92,266)( 93,263)( 94,264)( 95,262)( 96,261)( 97,260)( 98,259)
( 99,273)(100,274)(101,271)(102,272)(103,270)(104,269)(105,268)(106,267)
(107,281)(108,282)(109,279)(110,280)(111,278)(112,277)(113,276)(114,275)
(115,289)(116,290)(117,287)(118,288)(119,286)(120,285)(121,284)(122,283)
(123,297)(124,298)(125,295)(126,296)(127,294)(128,293)(129,292)(130,291)
(131,305)(132,306)(133,303)(134,304)(135,302)(136,301)(137,300)(138,299)
(139,313)(140,314)(141,311)(142,312)(143,310)(144,309)(145,308)(146,307)
(147,321)(148,322)(149,319)(150,320)(151,318)(152,317)(153,316)(154,315)
(155,329)(156,330)(157,327)(158,328)(159,326)(160,325)(161,324)(162,323)
(163,337)(164,338)(165,335)(166,336)(167,334)(168,333)(169,332)(170,331);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*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(338)!(1,2);
s1 := Sym(338)!(  5,  8)(  6,  7)(  9, 10)( 11, 51)( 12, 52)( 13, 56)( 14, 55)
( 15, 54)( 16, 53)( 17, 58)( 18, 57)( 19, 43)( 20, 44)( 21, 48)( 22, 47)
( 23, 46)( 24, 45)( 25, 50)( 26, 49)( 27, 35)( 28, 36)( 29, 40)( 30, 39)
( 31, 38)( 32, 37)( 33, 42)( 34, 41)( 59,115)( 60,116)( 61,120)( 62,119)
( 63,118)( 64,117)( 65,122)( 66,121)( 67,163)( 68,164)( 69,168)( 70,167)
( 71,166)( 72,165)( 73,170)( 74,169)( 75,155)( 76,156)( 77,160)( 78,159)
( 79,158)( 80,157)( 81,162)( 82,161)( 83,147)( 84,148)( 85,152)( 86,151)
( 87,150)( 88,149)( 89,154)( 90,153)( 91,139)( 92,140)( 93,144)( 94,143)
( 95,142)( 96,141)( 97,146)( 98,145)( 99,131)(100,132)(101,136)(102,135)
(103,134)(104,133)(105,138)(106,137)(107,123)(108,124)(109,128)(110,127)
(111,126)(112,125)(113,130)(114,129)(171,172)(173,175)(174,176)(179,220)
(180,219)(181,223)(182,224)(183,221)(184,222)(185,225)(186,226)(187,212)
(188,211)(189,215)(190,216)(191,213)(192,214)(193,217)(194,218)(195,204)
(196,203)(197,207)(198,208)(199,205)(200,206)(201,209)(202,210)(227,284)
(228,283)(229,287)(230,288)(231,285)(232,286)(233,289)(234,290)(235,332)
(236,331)(237,335)(238,336)(239,333)(240,334)(241,337)(242,338)(243,324)
(244,323)(245,327)(246,328)(247,325)(248,326)(249,329)(250,330)(251,316)
(252,315)(253,319)(254,320)(255,317)(256,318)(257,321)(258,322)(259,308)
(260,307)(261,311)(262,312)(263,309)(264,310)(265,313)(266,314)(267,300)
(268,299)(269,303)(270,304)(271,301)(272,302)(273,305)(274,306)(275,292)
(276,291)(277,295)(278,296)(279,293)(280,294)(281,297)(282,298);
s2 := Sym(338)!(  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 73)(  8, 74)(  9, 71)
( 10, 72)( 11, 59)( 12, 60)( 13, 62)( 14, 61)( 15, 65)( 16, 66)( 17, 63)
( 18, 64)( 19,107)( 20,108)( 21,110)( 22,109)( 23,113)( 24,114)( 25,111)
( 26,112)( 27, 99)( 28,100)( 29,102)( 30,101)( 31,105)( 32,106)( 33,103)
( 34,104)( 35, 91)( 36, 92)( 37, 94)( 38, 93)( 39, 97)( 40, 98)( 41, 95)
( 42, 96)( 43, 83)( 44, 84)( 45, 86)( 46, 85)( 47, 89)( 48, 90)( 49, 87)
( 50, 88)( 51, 75)( 52, 76)( 53, 78)( 54, 77)( 55, 81)( 56, 82)( 57, 79)
( 58, 80)(115,123)(116,124)(117,126)(118,125)(119,129)(120,130)(121,127)
(122,128)(131,163)(132,164)(133,166)(134,165)(135,169)(136,170)(137,167)
(138,168)(139,155)(140,156)(141,158)(142,157)(143,161)(144,162)(145,159)
(146,160)(149,150)(151,153)(152,154)(171,236)(172,235)(173,237)(174,238)
(175,242)(176,241)(177,240)(178,239)(179,228)(180,227)(181,229)(182,230)
(183,234)(184,233)(185,232)(186,231)(187,276)(188,275)(189,277)(190,278)
(191,282)(192,281)(193,280)(194,279)(195,268)(196,267)(197,269)(198,270)
(199,274)(200,273)(201,272)(202,271)(203,260)(204,259)(205,261)(206,262)
(207,266)(208,265)(209,264)(210,263)(211,252)(212,251)(213,253)(214,254)
(215,258)(216,257)(217,256)(218,255)(219,244)(220,243)(221,245)(222,246)
(223,250)(224,249)(225,248)(226,247)(283,292)(284,291)(285,293)(286,294)
(287,298)(288,297)(289,296)(290,295)(299,332)(300,331)(301,333)(302,334)
(303,338)(304,337)(305,336)(306,335)(307,324)(308,323)(309,325)(310,326)
(311,330)(312,329)(313,328)(314,327)(315,316)(319,322)(320,321);
s3 := Sym(338)!(  3,177)(  4,178)(  5,175)(  6,176)(  7,174)(  8,173)(  9,172)
( 10,171)( 11,185)( 12,186)( 13,183)( 14,184)( 15,182)( 16,181)( 17,180)
( 18,179)( 19,193)( 20,194)( 21,191)( 22,192)( 23,190)( 24,189)( 25,188)
( 26,187)( 27,201)( 28,202)( 29,199)( 30,200)( 31,198)( 32,197)( 33,196)
( 34,195)( 35,209)( 36,210)( 37,207)( 38,208)( 39,206)( 40,205)( 41,204)
( 42,203)( 43,217)( 44,218)( 45,215)( 46,216)( 47,214)( 48,213)( 49,212)
( 50,211)( 51,225)( 52,226)( 53,223)( 54,224)( 55,222)( 56,221)( 57,220)
( 58,219)( 59,233)( 60,234)( 61,231)( 62,232)( 63,230)( 64,229)( 65,228)
( 66,227)( 67,241)( 68,242)( 69,239)( 70,240)( 71,238)( 72,237)( 73,236)
( 74,235)( 75,249)( 76,250)( 77,247)( 78,248)( 79,246)( 80,245)( 81,244)
( 82,243)( 83,257)( 84,258)( 85,255)( 86,256)( 87,254)( 88,253)( 89,252)
( 90,251)( 91,265)( 92,266)( 93,263)( 94,264)( 95,262)( 96,261)( 97,260)
( 98,259)( 99,273)(100,274)(101,271)(102,272)(103,270)(104,269)(105,268)
(106,267)(107,281)(108,282)(109,279)(110,280)(111,278)(112,277)(113,276)
(114,275)(115,289)(116,290)(117,287)(118,288)(119,286)(120,285)(121,284)
(122,283)(123,297)(124,298)(125,295)(126,296)(127,294)(128,293)(129,292)
(130,291)(131,305)(132,306)(133,303)(134,304)(135,302)(136,301)(137,300)
(138,299)(139,313)(140,314)(141,311)(142,312)(143,310)(144,309)(145,308)
(146,307)(147,321)(148,322)(149,319)(150,320)(151,318)(152,317)(153,316)
(154,315)(155,329)(156,330)(157,327)(158,328)(159,326)(160,325)(161,324)
(162,323)(163,337)(164,338)(165,335)(166,336)(167,334)(168,333)(169,332)
(170,331);
poly := sub<Sym(338)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*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 >;

to this polytope