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Polytope of Type {84,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,6}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11355)
Rank : 3
Schlafli Type : {84,6}
Number of vertices, edges, etc : 112, 336, 8
Order of s0s1s2 : 56
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,6}*672
   4-fold quotients : {21,6}*336
   7-fold quotients : {12,6}*192b
   14-fold quotients : {12,3}*96, {6,6}*96
   24-fold quotients : {14,2}*56
   28-fold quotients : {3,6}*48, {6,3}*48
   48-fold quotients : {7,2}*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)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 55)
( 14, 56)( 15, 53)( 16, 54)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 47)
( 22, 48)( 23, 45)( 24, 46)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 29, 39)
( 30, 40)( 31, 37)( 32, 38)( 57,113)( 58,114)( 59,116)( 60,115)( 61,119)
( 62,120)( 63,117)( 64,118)( 65,161)( 66,162)( 67,164)( 68,163)( 69,167)
( 70,168)( 71,165)( 72,166)( 73,153)( 74,154)( 75,156)( 76,155)( 77,159)
( 78,160)( 79,157)( 80,158)( 81,145)( 82,146)( 83,148)( 84,147)( 85,151)
( 86,152)( 87,149)( 88,150)( 89,137)( 90,138)( 91,140)( 92,139)( 93,143)
( 94,144)( 95,141)( 96,142)( 97,129)( 98,130)( 99,132)(100,131)(101,135)
(102,136)(103,133)(104,134)(105,121)(106,122)(107,124)(108,123)(109,127)
(110,128)(111,125)(112,126)(169,170)(173,176)(174,175)(177,218)(178,217)
(179,219)(180,220)(181,224)(182,223)(183,222)(184,221)(185,210)(186,209)
(187,211)(188,212)(189,216)(190,215)(191,214)(192,213)(193,202)(194,201)
(195,203)(196,204)(197,208)(198,207)(199,206)(200,205)(225,282)(226,281)
(227,283)(228,284)(229,288)(230,287)(231,286)(232,285)(233,330)(234,329)
(235,331)(236,332)(237,336)(238,335)(239,334)(240,333)(241,322)(242,321)
(243,323)(244,324)(245,328)(246,327)(247,326)(248,325)(249,314)(250,313)
(251,315)(252,316)(253,320)(254,319)(255,318)(256,317)(257,306)(258,305)
(259,307)(260,308)(261,312)(262,311)(263,310)(264,309)(265,298)(266,297)
(267,299)(268,300)(269,304)(270,303)(271,302)(272,301)(273,290)(274,289)
(275,291)(276,292)(277,296)(278,295)(279,294)(280,293);;
s1 := (  1,233)(  2,234)(  3,239)(  4,240)(  5,238)(  6,237)(  7,235)(  8,236)
(  9,225)( 10,226)( 11,231)( 12,232)( 13,230)( 14,229)( 15,227)( 16,228)
( 17,273)( 18,274)( 19,279)( 20,280)( 21,278)( 22,277)( 23,275)( 24,276)
( 25,265)( 26,266)( 27,271)( 28,272)( 29,270)( 30,269)( 31,267)( 32,268)
( 33,257)( 34,258)( 35,263)( 36,264)( 37,262)( 38,261)( 39,259)( 40,260)
( 41,249)( 42,250)( 43,255)( 44,256)( 45,254)( 46,253)( 47,251)( 48,252)
( 49,241)( 50,242)( 51,247)( 52,248)( 53,246)( 54,245)( 55,243)( 56,244)
( 57,177)( 58,178)( 59,183)( 60,184)( 61,182)( 62,181)( 63,179)( 64,180)
( 65,169)( 66,170)( 67,175)( 68,176)( 69,174)( 70,173)( 71,171)( 72,172)
( 73,217)( 74,218)( 75,223)( 76,224)( 77,222)( 78,221)( 79,219)( 80,220)
( 81,209)( 82,210)( 83,215)( 84,216)( 85,214)( 86,213)( 87,211)( 88,212)
( 89,201)( 90,202)( 91,207)( 92,208)( 93,206)( 94,205)( 95,203)( 96,204)
( 97,193)( 98,194)( 99,199)(100,200)(101,198)(102,197)(103,195)(104,196)
(105,185)(106,186)(107,191)(108,192)(109,190)(110,189)(111,187)(112,188)
(113,289)(114,290)(115,295)(116,296)(117,294)(118,293)(119,291)(120,292)
(121,281)(122,282)(123,287)(124,288)(125,286)(126,285)(127,283)(128,284)
(129,329)(130,330)(131,335)(132,336)(133,334)(134,333)(135,331)(136,332)
(137,321)(138,322)(139,327)(140,328)(141,326)(142,325)(143,323)(144,324)
(145,313)(146,314)(147,319)(148,320)(149,318)(150,317)(151,315)(152,316)
(153,305)(154,306)(155,311)(156,312)(157,310)(158,309)(159,307)(160,308)
(161,297)(162,298)(163,303)(164,304)(165,302)(166,301)(167,299)(168,300);;
s2 := (  1,171)(  2,172)(  3,169)(  4,170)(  5,174)(  6,173)(  7,175)(  8,176)
(  9,179)( 10,180)( 11,177)( 12,178)( 13,182)( 14,181)( 15,183)( 16,184)
( 17,187)( 18,188)( 19,185)( 20,186)( 21,190)( 22,189)( 23,191)( 24,192)
( 25,195)( 26,196)( 27,193)( 28,194)( 29,198)( 30,197)( 31,199)( 32,200)
( 33,203)( 34,204)( 35,201)( 36,202)( 37,206)( 38,205)( 39,207)( 40,208)
( 41,211)( 42,212)( 43,209)( 44,210)( 45,214)( 46,213)( 47,215)( 48,216)
( 49,219)( 50,220)( 51,217)( 52,218)( 53,222)( 54,221)( 55,223)( 56,224)
( 57,283)( 58,284)( 59,281)( 60,282)( 61,286)( 62,285)( 63,287)( 64,288)
( 65,291)( 66,292)( 67,289)( 68,290)( 69,294)( 70,293)( 71,295)( 72,296)
( 73,299)( 74,300)( 75,297)( 76,298)( 77,302)( 78,301)( 79,303)( 80,304)
( 81,307)( 82,308)( 83,305)( 84,306)( 85,310)( 86,309)( 87,311)( 88,312)
( 89,315)( 90,316)( 91,313)( 92,314)( 93,318)( 94,317)( 95,319)( 96,320)
( 97,323)( 98,324)( 99,321)(100,322)(101,326)(102,325)(103,327)(104,328)
(105,331)(106,332)(107,329)(108,330)(109,334)(110,333)(111,335)(112,336)
(113,227)(114,228)(115,225)(116,226)(117,230)(118,229)(119,231)(120,232)
(121,235)(122,236)(123,233)(124,234)(125,238)(126,237)(127,239)(128,240)
(129,243)(130,244)(131,241)(132,242)(133,246)(134,245)(135,247)(136,248)
(137,251)(138,252)(139,249)(140,250)(141,254)(142,253)(143,255)(144,256)
(145,259)(146,260)(147,257)(148,258)(149,262)(150,261)(151,263)(152,264)
(153,267)(154,268)(155,265)(156,266)(157,270)(158,269)(159,271)(160,272)
(161,275)(162,276)(163,273)(164,274)(165,278)(166,277)(167,279)(168,280);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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,  4)(  5,  7)(  6,  8)(  9, 49)( 10, 50)( 11, 52)( 12, 51)
( 13, 55)( 14, 56)( 15, 53)( 16, 54)( 17, 41)( 18, 42)( 19, 44)( 20, 43)
( 21, 47)( 22, 48)( 23, 45)( 24, 46)( 25, 33)( 26, 34)( 27, 36)( 28, 35)
( 29, 39)( 30, 40)( 31, 37)( 32, 38)( 57,113)( 58,114)( 59,116)( 60,115)
( 61,119)( 62,120)( 63,117)( 64,118)( 65,161)( 66,162)( 67,164)( 68,163)
( 69,167)( 70,168)( 71,165)( 72,166)( 73,153)( 74,154)( 75,156)( 76,155)
( 77,159)( 78,160)( 79,157)( 80,158)( 81,145)( 82,146)( 83,148)( 84,147)
( 85,151)( 86,152)( 87,149)( 88,150)( 89,137)( 90,138)( 91,140)( 92,139)
( 93,143)( 94,144)( 95,141)( 96,142)( 97,129)( 98,130)( 99,132)(100,131)
(101,135)(102,136)(103,133)(104,134)(105,121)(106,122)(107,124)(108,123)
(109,127)(110,128)(111,125)(112,126)(169,170)(173,176)(174,175)(177,218)
(178,217)(179,219)(180,220)(181,224)(182,223)(183,222)(184,221)(185,210)
(186,209)(187,211)(188,212)(189,216)(190,215)(191,214)(192,213)(193,202)
(194,201)(195,203)(196,204)(197,208)(198,207)(199,206)(200,205)(225,282)
(226,281)(227,283)(228,284)(229,288)(230,287)(231,286)(232,285)(233,330)
(234,329)(235,331)(236,332)(237,336)(238,335)(239,334)(240,333)(241,322)
(242,321)(243,323)(244,324)(245,328)(246,327)(247,326)(248,325)(249,314)
(250,313)(251,315)(252,316)(253,320)(254,319)(255,318)(256,317)(257,306)
(258,305)(259,307)(260,308)(261,312)(262,311)(263,310)(264,309)(265,298)
(266,297)(267,299)(268,300)(269,304)(270,303)(271,302)(272,301)(273,290)
(274,289)(275,291)(276,292)(277,296)(278,295)(279,294)(280,293);
s1 := Sym(336)!(  1,233)(  2,234)(  3,239)(  4,240)(  5,238)(  6,237)(  7,235)
(  8,236)(  9,225)( 10,226)( 11,231)( 12,232)( 13,230)( 14,229)( 15,227)
( 16,228)( 17,273)( 18,274)( 19,279)( 20,280)( 21,278)( 22,277)( 23,275)
( 24,276)( 25,265)( 26,266)( 27,271)( 28,272)( 29,270)( 30,269)( 31,267)
( 32,268)( 33,257)( 34,258)( 35,263)( 36,264)( 37,262)( 38,261)( 39,259)
( 40,260)( 41,249)( 42,250)( 43,255)( 44,256)( 45,254)( 46,253)( 47,251)
( 48,252)( 49,241)( 50,242)( 51,247)( 52,248)( 53,246)( 54,245)( 55,243)
( 56,244)( 57,177)( 58,178)( 59,183)( 60,184)( 61,182)( 62,181)( 63,179)
( 64,180)( 65,169)( 66,170)( 67,175)( 68,176)( 69,174)( 70,173)( 71,171)
( 72,172)( 73,217)( 74,218)( 75,223)( 76,224)( 77,222)( 78,221)( 79,219)
( 80,220)( 81,209)( 82,210)( 83,215)( 84,216)( 85,214)( 86,213)( 87,211)
( 88,212)( 89,201)( 90,202)( 91,207)( 92,208)( 93,206)( 94,205)( 95,203)
( 96,204)( 97,193)( 98,194)( 99,199)(100,200)(101,198)(102,197)(103,195)
(104,196)(105,185)(106,186)(107,191)(108,192)(109,190)(110,189)(111,187)
(112,188)(113,289)(114,290)(115,295)(116,296)(117,294)(118,293)(119,291)
(120,292)(121,281)(122,282)(123,287)(124,288)(125,286)(126,285)(127,283)
(128,284)(129,329)(130,330)(131,335)(132,336)(133,334)(134,333)(135,331)
(136,332)(137,321)(138,322)(139,327)(140,328)(141,326)(142,325)(143,323)
(144,324)(145,313)(146,314)(147,319)(148,320)(149,318)(150,317)(151,315)
(152,316)(153,305)(154,306)(155,311)(156,312)(157,310)(158,309)(159,307)
(160,308)(161,297)(162,298)(163,303)(164,304)(165,302)(166,301)(167,299)
(168,300);
s2 := Sym(336)!(  1,171)(  2,172)(  3,169)(  4,170)(  5,174)(  6,173)(  7,175)
(  8,176)(  9,179)( 10,180)( 11,177)( 12,178)( 13,182)( 14,181)( 15,183)
( 16,184)( 17,187)( 18,188)( 19,185)( 20,186)( 21,190)( 22,189)( 23,191)
( 24,192)( 25,195)( 26,196)( 27,193)( 28,194)( 29,198)( 30,197)( 31,199)
( 32,200)( 33,203)( 34,204)( 35,201)( 36,202)( 37,206)( 38,205)( 39,207)
( 40,208)( 41,211)( 42,212)( 43,209)( 44,210)( 45,214)( 46,213)( 47,215)
( 48,216)( 49,219)( 50,220)( 51,217)( 52,218)( 53,222)( 54,221)( 55,223)
( 56,224)( 57,283)( 58,284)( 59,281)( 60,282)( 61,286)( 62,285)( 63,287)
( 64,288)( 65,291)( 66,292)( 67,289)( 68,290)( 69,294)( 70,293)( 71,295)
( 72,296)( 73,299)( 74,300)( 75,297)( 76,298)( 77,302)( 78,301)( 79,303)
( 80,304)( 81,307)( 82,308)( 83,305)( 84,306)( 85,310)( 86,309)( 87,311)
( 88,312)( 89,315)( 90,316)( 91,313)( 92,314)( 93,318)( 94,317)( 95,319)
( 96,320)( 97,323)( 98,324)( 99,321)(100,322)(101,326)(102,325)(103,327)
(104,328)(105,331)(106,332)(107,329)(108,330)(109,334)(110,333)(111,335)
(112,336)(113,227)(114,228)(115,225)(116,226)(117,230)(118,229)(119,231)
(120,232)(121,235)(122,236)(123,233)(124,234)(125,238)(126,237)(127,239)
(128,240)(129,243)(130,244)(131,241)(132,242)(133,246)(134,245)(135,247)
(136,248)(137,251)(138,252)(139,249)(140,250)(141,254)(142,253)(143,255)
(144,256)(145,259)(146,260)(147,257)(148,258)(149,262)(150,261)(151,263)
(152,264)(153,267)(154,268)(155,265)(156,266)(157,270)(158,269)(159,271)
(160,272)(161,275)(162,276)(163,273)(164,274)(165,278)(166,277)(167,279)
(168,280);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope