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Polytope of Type {6,112}

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