Polytope of Type {6,112}

Play with this polytope as a twisty puzzle

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.
Irregular Quotients (of which this is a minimal cover):
   None.

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

Twisty Puzzle