Polytope of Type {2,8,21}

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

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

Permutation Representation (Magma) :

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

to this polytope