Polytope of Type {84,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,6}*1344a
if this polytope has a name.
Group : SmallGroup(1344,11328)
Rank : 3
Schlafli Type : {84,6}
Number of vertices, edges, etc : 112, 336, 8
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 6
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}*192a
   12-fold quotients : {28,2}*112
   14-fold quotients : {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
   84-fold quotients : {4,2}*16
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope