Polytope of Type {168,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {168,4}*1344a
Also Known As : {168,4|2}. if this polytope has another name.
Group : SmallGroup(1344,5658)
Rank : 3
Schlafli Type : {168,4}
Number of vertices, edges, etc : 168, 336, 4
Order of s0s1s2 : 168
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {84,4}*672a, {168,2}*672
   3-fold quotients : {56,4}*448a
   4-fold quotients : {84,2}*336, {42,4}*336a
   6-fold quotients : {28,4}*224, {56,2}*224
   7-fold quotients : {24,4}*192a
   8-fold quotients : {42,2}*168
   12-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {12,4}*96a, {24,2}*96
   16-fold quotients : {21,2}*84
   21-fold quotients : {8,4}*64a
   24-fold quotients : {14,2}*56
   28-fold quotients : {12,2}*48, {6,4}*48a
   42-fold quotients : {4,4}*32, {8,2}*32
   48-fold quotients : {7,2}*28
   56-fold quotients : {6,2}*24
   84-fold quotients : {2,4}*16, {4,2}*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 := (  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 64)( 44, 70)( 45, 69)( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 78)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 79)( 57, 71)( 58, 77)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)( 96,102)( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(127,148)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,162)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,155)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(169,211)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,225)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,218)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219)(190,232)(191,238)(192,237)(193,236)(194,235)(195,234)(196,233)(197,246)(198,252)(199,251)(200,250)(201,249)(202,248)(203,247)(204,239)(205,245)(206,244)(207,243)(208,242)(209,241)(210,240)(253,295)(254,301)(255,300)(256,299)(257,298)(258,297)(259,296)(260,309)(261,315)(262,314)(263,313)(264,312)(265,311)(266,310)(267,302)(268,308)(269,307)(270,306)(271,305)(272,304)(273,303)(274,316)(275,322)(276,321)(277,320)(278,319)(279,318)(280,317)(281,330)(282,336)(283,335)(284,334)(285,333)(286,332)(287,331)(288,323)(289,329)(290,328)(291,327)(292,326)(293,325)(294,324);;
s1 := (  1,177)(  2,176)(  3,182)(  4,181)(  5,180)(  6,179)(  7,178)(  8,170)(  9,169)( 10,175)( 11,174)( 12,173)( 13,172)( 14,171)( 15,184)( 16,183)( 17,189)( 18,188)( 19,187)( 20,186)( 21,185)( 22,198)( 23,197)( 24,203)( 25,202)( 26,201)( 27,200)( 28,199)( 29,191)( 30,190)( 31,196)( 32,195)( 33,194)( 34,193)( 35,192)( 36,205)( 37,204)( 38,210)( 39,209)( 40,208)( 41,207)( 42,206)( 43,240)( 44,239)( 45,245)( 46,244)( 47,243)( 48,242)( 49,241)( 50,233)( 51,232)( 52,238)( 53,237)( 54,236)( 55,235)( 56,234)( 57,247)( 58,246)( 59,252)( 60,251)( 61,250)( 62,249)( 63,248)( 64,219)( 65,218)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,212)( 72,211)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,226)( 79,225)( 80,231)( 81,230)( 82,229)( 83,228)( 84,227)( 85,261)( 86,260)( 87,266)( 88,265)( 89,264)( 90,263)( 91,262)( 92,254)( 93,253)( 94,259)( 95,258)( 96,257)( 97,256)( 98,255)( 99,268)(100,267)(101,273)(102,272)(103,271)(104,270)(105,269)(106,282)(107,281)(108,287)(109,286)(110,285)(111,284)(112,283)(113,275)(114,274)(115,280)(116,279)(117,278)(118,277)(119,276)(120,289)(121,288)(122,294)(123,293)(124,292)(125,291)(126,290)(127,324)(128,323)(129,329)(130,328)(131,327)(132,326)(133,325)(134,317)(135,316)(136,322)(137,321)(138,320)(139,319)(140,318)(141,331)(142,330)(143,336)(144,335)(145,334)(146,333)(147,332)(148,303)(149,302)(150,308)(151,307)(152,306)(153,305)(154,304)(155,296)(156,295)(157,301)(158,300)(159,299)(160,298)(161,297)(162,310)(163,309)(164,315)(165,314)(166,313)(167,312)(168,311);;
s2 := (169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)(249,333)(250,334)(251,335)(252,336);;
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*s2*s1, 
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*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)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 64)( 44, 70)( 45, 69)( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 78)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 79)( 57, 71)( 58, 77)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)( 96,102)( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(127,148)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,162)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,155)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(169,211)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,225)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,218)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219)(190,232)(191,238)(192,237)(193,236)(194,235)(195,234)(196,233)(197,246)(198,252)(199,251)(200,250)(201,249)(202,248)(203,247)(204,239)(205,245)(206,244)(207,243)(208,242)(209,241)(210,240)(253,295)(254,301)(255,300)(256,299)(257,298)(258,297)(259,296)(260,309)(261,315)(262,314)(263,313)(264,312)(265,311)(266,310)(267,302)(268,308)(269,307)(270,306)(271,305)(272,304)(273,303)(274,316)(275,322)(276,321)(277,320)(278,319)(279,318)(280,317)(281,330)(282,336)(283,335)(284,334)(285,333)(286,332)(287,331)(288,323)(289,329)(290,328)(291,327)(292,326)(293,325)(294,324);
s1 := Sym(336)!(  1,177)(  2,176)(  3,182)(  4,181)(  5,180)(  6,179)(  7,178)(  8,170)(  9,169)( 10,175)( 11,174)( 12,173)( 13,172)( 14,171)( 15,184)( 16,183)( 17,189)( 18,188)( 19,187)( 20,186)( 21,185)( 22,198)( 23,197)( 24,203)( 25,202)( 26,201)( 27,200)( 28,199)( 29,191)( 30,190)( 31,196)( 32,195)( 33,194)( 34,193)( 35,192)( 36,205)( 37,204)( 38,210)( 39,209)( 40,208)( 41,207)( 42,206)( 43,240)( 44,239)( 45,245)( 46,244)( 47,243)( 48,242)( 49,241)( 50,233)( 51,232)( 52,238)( 53,237)( 54,236)( 55,235)( 56,234)( 57,247)( 58,246)( 59,252)( 60,251)( 61,250)( 62,249)( 63,248)( 64,219)( 65,218)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,212)( 72,211)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,226)( 79,225)( 80,231)( 81,230)( 82,229)( 83,228)( 84,227)( 85,261)( 86,260)( 87,266)( 88,265)( 89,264)( 90,263)( 91,262)( 92,254)( 93,253)( 94,259)( 95,258)( 96,257)( 97,256)( 98,255)( 99,268)(100,267)(101,273)(102,272)(103,271)(104,270)(105,269)(106,282)(107,281)(108,287)(109,286)(110,285)(111,284)(112,283)(113,275)(114,274)(115,280)(116,279)(117,278)(118,277)(119,276)(120,289)(121,288)(122,294)(123,293)(124,292)(125,291)(126,290)(127,324)(128,323)(129,329)(130,328)(131,327)(132,326)(133,325)(134,317)(135,316)(136,322)(137,321)(138,320)(139,319)(140,318)(141,331)(142,330)(143,336)(144,335)(145,334)(146,333)(147,332)(148,303)(149,302)(150,308)(151,307)(152,306)(153,305)(154,304)(155,296)(156,295)(157,301)(158,300)(159,299)(160,298)(161,297)(162,310)(163,309)(164,315)(165,314)(166,313)(167,312)(168,311);
s2 := Sym(336)!(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)(249,333)(250,334)(251,335)(252,336);
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*s2*s1, 
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*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

Twisty Puzzle