Polytope of Type {42,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,4}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11399)
Rank : 3
Schlafli Type : {42,4}
Number of vertices, edges, etc : 168, 336, 16
Order of s0s1s2 : 84
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {42,4}*672
   4-fold quotients : {42,4}*336a, {21,4}*336, {42,4}*336b, {42,4}*336c
   7-fold quotients : {6,4}*192b
   8-fold quotients : {21,4}*168, {42,2}*168
   12-fold quotients : {14,4}*112
   14-fold quotients : {6,4}*96
   16-fold quotients : {21,2}*84
   24-fold quotients : {14,2}*56
   28-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {7,2}*28
   56-fold quotients : {3,4}*24, {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):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      8 facets:
         8 of {42}*84
      84 vertex figures:
         84 of {4}*8
   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> of order 2.
      12 facets:
         8 of {21}*42
         4 of {42}*84
      84 vertex figures:
         84 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      8 facets:
         8 of {42}*84
      84 vertex figures:
         84 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2.
      8 facets:
         8 of {42}*84
      112 vertex figures:
         56 of {4}*8
         56 of {2}*4
   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> of order 3.
      8 facets:
         4 of {14}*28
         4 of {42}*84
      56 vertex figures:
         56 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*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> of order 4.
      6 facets:
         4 of {21}*42
         2 of {42}*84
      56 vertex figures:
         28 of {4}*8
         28 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
      4 facets:
         4 of {42}*84
      70 vertex figures:
         56 of {2}*4
         14 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      4 facets:
         4 of {42}*84
      42 vertex figures:
         42 of {4}*8

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

Twisty Puzzle