Polytope of Type {84,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,4}*1344a
if this polytope has a name.
Group : SmallGroup(1344,5667)
Rank : 3
Schlafli Type : {84,4}
Number of vertices, edges, etc : 168, 336, 8
Order of s0s1s2 : 84
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   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
   3-fold quotients : {28,4}*448
   4-fold quotients : {84,2}*336, {42,4}*336a
   6-fold quotients : {28,4}*224
   7-fold quotients : {12,4}*192a
   8-fold quotients : {42,2}*168
   12-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {12,4}*96a
   16-fold quotients : {21,2}*84
   21-fold quotients : {4,4}*64
   24-fold quotients : {14,2}*56
   28-fold quotients : {12,2}*48, {6,4}*48a
   42-fold quotients : {4,4}*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):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      6 facets:
         4 of {42}*84
         2 of {84}*168
      84 vertex figures:
         84 of {4}*8
   P/N, where N=<s1*s2*s1*s2> of order 2.
      4 facets:
         4 of {84}*168
      126 vertex figures:
         84 of {2}*4
         42 of {4}*8

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)( 44, 49)( 45, 48)( 46, 47)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(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)(253,316)(254,322)(255,321)(256,320)(257,319)(258,318)(259,317)(260,330)(261,336)(262,335)(263,334)(264,333)(265,332)(266,331)(267,323)(268,329)(269,328)(270,327)(271,326)(272,325)(273,324)(274,295)(275,301)(276,300)(277,299)(278,298)(279,297)(280,296)(281,309)(282,315)(283,314)(284,313)(285,312)(286,311)(287,310)(288,302)(289,308)(290,307)(291,306)(292,305)(293,304)(294,303);;
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)( 43,219)( 44,218)( 45,224)( 46,223)( 47,222)( 48,221)( 49,220)( 50,212)( 51,211)( 52,217)( 53,216)( 54,215)( 55,214)( 56,213)( 57,226)( 58,225)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)( 64,240)( 65,239)( 66,245)( 67,244)( 68,243)( 69,242)( 70,241)( 71,233)( 72,232)( 73,238)( 74,237)( 75,236)( 76,235)( 77,234)( 78,247)( 79,246)( 80,252)( 81,251)( 82,250)( 83,249)( 84,248)( 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)(127,303)(128,302)(129,308)(130,307)(131,306)(132,305)(133,304)(134,296)(135,295)(136,301)(137,300)(138,299)(139,298)(140,297)(141,310)(142,309)(143,315)(144,314)(145,313)(146,312)(147,311)(148,324)(149,323)(150,329)(151,328)(152,327)(153,326)(154,325)(155,317)(156,316)(157,322)(158,321)(159,320)(160,319)(161,318)(162,331)(163,330)(164,336)(165,335)(166,334)(167,333)(168,332);;
s2 := ( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 71)( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)( 58, 79)( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(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)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,330)(226,331)(227,332)(228,333)(229,334)(230,335)(231,336)(232,295)(233,296)(234,297)(235,298)(236,299)(237,300)(238,301)(239,302)(240,303)(241,304)(242,305)(243,306)(244,307)(245,308)(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 44, 49)( 45, 48)( 46, 47)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(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)(253,316)(254,322)(255,321)(256,320)(257,319)(258,318)(259,317)(260,330)(261,336)(262,335)(263,334)(264,333)(265,332)(266,331)(267,323)(268,329)(269,328)(270,327)(271,326)(272,325)(273,324)(274,295)(275,301)(276,300)(277,299)(278,298)(279,297)(280,296)(281,309)(282,315)(283,314)(284,313)(285,312)(286,311)(287,310)(288,302)(289,308)(290,307)(291,306)(292,305)(293,304)(294,303);
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)( 43,219)( 44,218)( 45,224)( 46,223)( 47,222)( 48,221)( 49,220)( 50,212)( 51,211)( 52,217)( 53,216)( 54,215)( 55,214)( 56,213)( 57,226)( 58,225)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)( 64,240)( 65,239)( 66,245)( 67,244)( 68,243)( 69,242)( 70,241)( 71,233)( 72,232)( 73,238)( 74,237)( 75,236)( 76,235)( 77,234)( 78,247)( 79,246)( 80,252)( 81,251)( 82,250)( 83,249)( 84,248)( 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)(127,303)(128,302)(129,308)(130,307)(131,306)(132,305)(133,304)(134,296)(135,295)(136,301)(137,300)(138,299)(139,298)(140,297)(141,310)(142,309)(143,315)(144,314)(145,313)(146,312)(147,311)(148,324)(149,323)(150,329)(151,328)(152,327)(153,326)(154,325)(155,317)(156,316)(157,322)(158,321)(159,320)(160,319)(161,318)(162,331)(163,330)(164,336)(165,335)(166,334)(167,333)(168,332);
s2 := Sym(336)!( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 71)( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)( 58, 79)( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(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)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,330)(226,331)(227,332)(228,333)(229,334)(230,335)(231,336)(232,295)(233,296)(234,297)(235,298)(236,299)(237,300)(238,301)(239,302)(240,303)(241,304)(242,305)(243,306)(244,307)(245,308)(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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