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Polytope of Type {112,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {112,6}*1344
Also Known As : {112,6|2}. if this polytope has another name.
Group : SmallGroup(1344,1488)
Rank : 3
Schlafli Type : {112,6}
Number of vertices, edges, etc : 112, 336, 6
Order of s0s1s2 : 336
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {56,6}*672
   3-fold quotients : {112,2}*448
   4-fold quotients : {28,6}*336a
   6-fold quotients : {56,2}*224
   7-fold quotients : {16,6}*192
   8-fold quotients : {14,6}*168
   12-fold quotients : {28,2}*112
   14-fold quotients : {8,6}*96
   21-fold quotients : {16,2}*64
   24-fold quotients : {14,2}*56
   28-fold quotients : {4,6}*48a
   42-fold quotients : {8,2}*32
   48-fold quotients : {7,2}*28
   56-fold quotients : {2,6}*24
   84-fold quotients : {4,2}*16
   112-fold quotients : {2,3}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)( 47, 67)( 48, 66)
( 49, 65)( 50, 71)( 51, 77)( 52, 76)( 53, 75)( 54, 74)( 55, 73)( 56, 72)
( 57, 78)( 58, 84)( 59, 83)( 60, 82)( 61, 81)( 62, 80)( 63, 79)( 85,127)
( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)
( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,141)(100,147)(101,146)
(102,145)(103,144)(104,143)(105,142)(106,148)(107,154)(108,153)(109,152)
(110,151)(111,150)(112,149)(113,155)(114,161)(115,160)(116,159)(117,158)
(118,157)(119,156)(120,162)(121,168)(122,167)(123,166)(124,165)(125,164)
(126,163)(169,253)(170,259)(171,258)(172,257)(173,256)(174,255)(175,254)
(176,260)(177,266)(178,265)(179,264)(180,263)(181,262)(182,261)(183,267)
(184,273)(185,272)(186,271)(187,270)(188,269)(189,268)(190,274)(191,280)
(192,279)(193,278)(194,277)(195,276)(196,275)(197,281)(198,287)(199,286)
(200,285)(201,284)(202,283)(203,282)(204,288)(205,294)(206,293)(207,292)
(208,291)(209,290)(210,289)(211,316)(212,322)(213,321)(214,320)(215,319)
(216,318)(217,317)(218,323)(219,329)(220,328)(221,327)(222,326)(223,325)
(224,324)(225,330)(226,336)(227,335)(228,334)(229,333)(230,332)(231,331)
(232,295)(233,301)(234,300)(235,299)(236,298)(237,297)(238,296)(239,302)
(240,308)(241,307)(242,306)(243,305)(244,304)(245,303)(246,309)(247,315)
(248,314)(249,313)(250,312)(251,311)(252,310);;
s1 := (  1,170)(  2,169)(  3,175)(  4,174)(  5,173)(  6,172)(  7,171)(  8,184)
(  9,183)( 10,189)( 11,188)( 12,187)( 13,186)( 14,185)( 15,177)( 16,176)
( 17,182)( 18,181)( 19,180)( 20,179)( 21,178)( 22,191)( 23,190)( 24,196)
( 25,195)( 26,194)( 27,193)( 28,192)( 29,205)( 30,204)( 31,210)( 32,209)
( 33,208)( 34,207)( 35,206)( 36,198)( 37,197)( 38,203)( 39,202)( 40,201)
( 41,200)( 42,199)( 43,233)( 44,232)( 45,238)( 46,237)( 47,236)( 48,235)
( 49,234)( 50,247)( 51,246)( 52,252)( 53,251)( 54,250)( 55,249)( 56,248)
( 57,240)( 58,239)( 59,245)( 60,244)( 61,243)( 62,242)( 63,241)( 64,212)
( 65,211)( 66,217)( 67,216)( 68,215)( 69,214)( 70,213)( 71,226)( 72,225)
( 73,231)( 74,230)( 75,229)( 76,228)( 77,227)( 78,219)( 79,218)( 80,224)
( 81,223)( 82,222)( 83,221)( 84,220)( 85,296)( 86,295)( 87,301)( 88,300)
( 89,299)( 90,298)( 91,297)( 92,310)( 93,309)( 94,315)( 95,314)( 96,313)
( 97,312)( 98,311)( 99,303)(100,302)(101,308)(102,307)(103,306)(104,305)
(105,304)(106,317)(107,316)(108,322)(109,321)(110,320)(111,319)(112,318)
(113,331)(114,330)(115,336)(116,335)(117,334)(118,333)(119,332)(120,324)
(121,323)(122,329)(123,328)(124,327)(125,326)(126,325)(127,254)(128,253)
(129,259)(130,258)(131,257)(132,256)(133,255)(134,268)(135,267)(136,273)
(137,272)(138,271)(139,270)(140,269)(141,261)(142,260)(143,266)(144,265)
(145,264)(146,263)(147,262)(148,275)(149,274)(150,280)(151,279)(152,278)
(153,277)(154,276)(155,289)(156,288)(157,294)(158,293)(159,292)(160,291)
(161,290)(162,282)(163,281)(164,287)(165,286)(166,285)(167,284)(168,283);;
s2 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 22, 29)
( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)( 88, 95)
( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)(110,117)
(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)
(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)
(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,197)
(191,198)(192,199)(193,200)(194,201)(195,202)(196,203)(211,218)(212,219)
(213,220)(214,221)(215,222)(216,223)(217,224)(232,239)(233,240)(234,241)
(235,242)(236,243)(237,244)(238,245)(253,260)(254,261)(255,262)(256,263)
(257,264)(258,265)(259,266)(274,281)(275,282)(276,283)(277,284)(278,285)
(279,286)(280,287)(295,302)(296,303)(297,304)(298,305)(299,306)(300,307)
(301,308)(316,323)(317,324)(318,325)(319,326)(320,327)(321,328)(322,329);;
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*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*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*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*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*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*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(336)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)( 47, 67)
( 48, 66)( 49, 65)( 50, 71)( 51, 77)( 52, 76)( 53, 75)( 54, 74)( 55, 73)
( 56, 72)( 57, 78)( 58, 84)( 59, 83)( 60, 82)( 61, 81)( 62, 80)( 63, 79)
( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)
( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,141)(100,147)
(101,146)(102,145)(103,144)(104,143)(105,142)(106,148)(107,154)(108,153)
(109,152)(110,151)(111,150)(112,149)(113,155)(114,161)(115,160)(116,159)
(117,158)(118,157)(119,156)(120,162)(121,168)(122,167)(123,166)(124,165)
(125,164)(126,163)(169,253)(170,259)(171,258)(172,257)(173,256)(174,255)
(175,254)(176,260)(177,266)(178,265)(179,264)(180,263)(181,262)(182,261)
(183,267)(184,273)(185,272)(186,271)(187,270)(188,269)(189,268)(190,274)
(191,280)(192,279)(193,278)(194,277)(195,276)(196,275)(197,281)(198,287)
(199,286)(200,285)(201,284)(202,283)(203,282)(204,288)(205,294)(206,293)
(207,292)(208,291)(209,290)(210,289)(211,316)(212,322)(213,321)(214,320)
(215,319)(216,318)(217,317)(218,323)(219,329)(220,328)(221,327)(222,326)
(223,325)(224,324)(225,330)(226,336)(227,335)(228,334)(229,333)(230,332)
(231,331)(232,295)(233,301)(234,300)(235,299)(236,298)(237,297)(238,296)
(239,302)(240,308)(241,307)(242,306)(243,305)(244,304)(245,303)(246,309)
(247,315)(248,314)(249,313)(250,312)(251,311)(252,310);
s1 := Sym(336)!(  1,170)(  2,169)(  3,175)(  4,174)(  5,173)(  6,172)(  7,171)
(  8,184)(  9,183)( 10,189)( 11,188)( 12,187)( 13,186)( 14,185)( 15,177)
( 16,176)( 17,182)( 18,181)( 19,180)( 20,179)( 21,178)( 22,191)( 23,190)
( 24,196)( 25,195)( 26,194)( 27,193)( 28,192)( 29,205)( 30,204)( 31,210)
( 32,209)( 33,208)( 34,207)( 35,206)( 36,198)( 37,197)( 38,203)( 39,202)
( 40,201)( 41,200)( 42,199)( 43,233)( 44,232)( 45,238)( 46,237)( 47,236)
( 48,235)( 49,234)( 50,247)( 51,246)( 52,252)( 53,251)( 54,250)( 55,249)
( 56,248)( 57,240)( 58,239)( 59,245)( 60,244)( 61,243)( 62,242)( 63,241)
( 64,212)( 65,211)( 66,217)( 67,216)( 68,215)( 69,214)( 70,213)( 71,226)
( 72,225)( 73,231)( 74,230)( 75,229)( 76,228)( 77,227)( 78,219)( 79,218)
( 80,224)( 81,223)( 82,222)( 83,221)( 84,220)( 85,296)( 86,295)( 87,301)
( 88,300)( 89,299)( 90,298)( 91,297)( 92,310)( 93,309)( 94,315)( 95,314)
( 96,313)( 97,312)( 98,311)( 99,303)(100,302)(101,308)(102,307)(103,306)
(104,305)(105,304)(106,317)(107,316)(108,322)(109,321)(110,320)(111,319)
(112,318)(113,331)(114,330)(115,336)(116,335)(117,334)(118,333)(119,332)
(120,324)(121,323)(122,329)(123,328)(124,327)(125,326)(126,325)(127,254)
(128,253)(129,259)(130,258)(131,257)(132,256)(133,255)(134,268)(135,267)
(136,273)(137,272)(138,271)(139,270)(140,269)(141,261)(142,260)(143,266)
(144,265)(145,264)(146,263)(147,262)(148,275)(149,274)(150,280)(151,279)
(152,278)(153,277)(154,276)(155,289)(156,288)(157,294)(158,293)(159,292)
(160,291)(161,290)(162,282)(163,281)(164,287)(165,286)(166,285)(167,284)
(168,283);
s2 := Sym(336)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 22, 29)( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)
( 88, 95)( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)
(110,117)(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)
(132,139)(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)
(154,161)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)
(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203)(211,218)
(212,219)(213,220)(214,221)(215,222)(216,223)(217,224)(232,239)(233,240)
(234,241)(235,242)(236,243)(237,244)(238,245)(253,260)(254,261)(255,262)
(256,263)(257,264)(258,265)(259,266)(274,281)(275,282)(276,283)(277,284)
(278,285)(279,286)(280,287)(295,302)(296,303)(297,304)(298,305)(299,306)
(300,307)(301,308)(316,323)(317,324)(318,325)(319,326)(320,327)(321,328)
(322,329);
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*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*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*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*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*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*s1*s0*s1 >; 
 
References : None.
to this polytope