Polytope of Type {6,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,84}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11355)
Rank : 3
Schlafli Type : {6,84}
Number of vertices, edges, etc : 8, 336, 112
Order of s₀s₁s₂ : 56
Order of s₀s₁s₂s₁ : 12
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 : {6,42}*672
   4-fold quotients : {6,21}*336
   7-fold quotients : {6,12}*192b
   14-fold quotients : {3,12}*96, {6,6}*96
   24-fold quotients : {2,14}*56
   28-fold quotients : {3,6}*48, {6,3}*48
   48-fold quotients : {2,7}*28
   56-fold quotients : {3,3}*24
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope