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