Polytope of Type {4,42}

Play with this polytope as a twisty puzzle

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

Permutation Representation (GAP) : 
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,255)(170,256)(171,253)(172,254)(173,259)(174,260)(175,257)(176,258)(177,263)(178,264)(179,261)(180,262)(181,267)(182,268)(183,265)(184,266)(185,271)(186,272)(187,269)(188,270)(189,275)(190,276)(191,273)(192,274)(193,279)(194,280)(195,277)(196,278)(197,283)(198,284)(199,281)(200,282)(201,287)(202,288)(203,285)(204,286)(205,291)(206,292)(207,289)(208,290)(209,295)(210,296)(211,293)(212,294)(213,299)(214,300)(215,297)(216,298)(217,303)(218,304)(219,301)(220,302)(221,307)(222,308)(223,305)(224,306)(225,311)(226,312)(227,309)(228,310)(229,315)(230,316)(231,313)(232,314)(233,319)(234,320)(235,317)(236,318)(237,323)(238,324)(239,321)(240,322)(241,327)(242,328)(243,325)(244,326)(245,331)(246,332)(247,329)(248,330)(249,335)(250,336)(251,333)(252,334);;
s1 := (  1,169)(  2,170)(  3,172)(  4,171)(  5,193)(  6,194)(  7,196)(  8,195)(  9,189)( 10,190)( 11,192)( 12,191)( 13,185)( 14,186)( 15,188)( 16,187)( 17,181)( 18,182)( 19,184)( 20,183)( 21,177)( 22,178)( 23,180)( 24,179)( 25,173)( 26,174)( 27,176)( 28,175)( 29,225)( 30,226)( 31,228)( 32,227)( 33,249)( 34,250)( 35,252)( 36,251)( 37,245)( 38,246)( 39,248)( 40,247)( 41,241)( 42,242)( 43,244)( 44,243)( 45,237)( 46,238)( 47,240)( 48,239)( 49,233)( 50,234)( 51,236)( 52,235)( 53,229)( 54,230)( 55,232)( 56,231)( 57,197)( 58,198)( 59,200)( 60,199)( 61,221)( 62,222)( 63,224)( 64,223)( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)( 72,215)( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)( 80,207)( 81,201)( 82,202)( 83,204)( 84,203)( 85,253)( 86,254)( 87,256)( 88,255)( 89,277)( 90,278)( 91,280)( 92,279)( 93,273)( 94,274)( 95,276)( 96,275)( 97,269)( 98,270)( 99,272)(100,271)(101,265)(102,266)(103,268)(104,267)(105,261)(106,262)(107,264)(108,263)(109,257)(110,258)(111,260)(112,259)(113,309)(114,310)(115,312)(116,311)(117,333)(118,334)(119,336)(120,335)(121,329)(122,330)(123,332)(124,331)(125,325)(126,326)(127,328)(128,327)(129,321)(130,322)(131,324)(132,323)(133,317)(134,318)(135,320)(136,319)(137,313)(138,314)(139,316)(140,315)(141,281)(142,282)(143,284)(144,283)(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)(151,304)(152,303)(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)(159,296)(160,295)(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)(167,288)(168,287);;
s2 := (  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)(169,201)(170,204)(171,203)(172,202)(173,197)(174,200)(175,199)(176,198)(177,221)(178,224)(179,223)(180,222)(181,217)(182,220)(183,219)(184,218)(185,213)(186,216)(187,215)(188,214)(189,209)(190,212)(191,211)(192,210)(193,205)(194,208)(195,207)(196,206)(225,229)(226,232)(227,231)(228,230)(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)(239,247)(240,246)(242,244)(253,285)(254,288)(255,287)(256,286)(257,281)(258,284)(259,283)(260,282)(261,305)(262,308)(263,307)(264,306)(265,301)(266,304)(267,303)(268,302)(269,297)(270,300)(271,299)(272,298)(273,293)(274,296)(275,295)(276,294)(277,289)(278,292)(279,291)(280,290)(309,313)(310,316)(311,315)(312,314)(317,333)(318,336)(319,335)(320,334)(321,329)(322,332)(323,331)(324,330)(326,328);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,255)(170,256)(171,253)(172,254)(173,259)(174,260)(175,257)(176,258)(177,263)(178,264)(179,261)(180,262)(181,267)(182,268)(183,265)(184,266)(185,271)(186,272)(187,269)(188,270)(189,275)(190,276)(191,273)(192,274)(193,279)(194,280)(195,277)(196,278)(197,283)(198,284)(199,281)(200,282)(201,287)(202,288)(203,285)(204,286)(205,291)(206,292)(207,289)(208,290)(209,295)(210,296)(211,293)(212,294)(213,299)(214,300)(215,297)(216,298)(217,303)(218,304)(219,301)(220,302)(221,307)(222,308)(223,305)(224,306)(225,311)(226,312)(227,309)(228,310)(229,315)(230,316)(231,313)(232,314)(233,319)(234,320)(235,317)(236,318)(237,323)(238,324)(239,321)(240,322)(241,327)(242,328)(243,325)(244,326)(245,331)(246,332)(247,329)(248,330)(249,335)(250,336)(251,333)(252,334);
s1 := Sym(336)!(  1,169)(  2,170)(  3,172)(  4,171)(  5,193)(  6,194)(  7,196)(  8,195)(  9,189)( 10,190)( 11,192)( 12,191)( 13,185)( 14,186)( 15,188)( 16,187)( 17,181)( 18,182)( 19,184)( 20,183)( 21,177)( 22,178)( 23,180)( 24,179)( 25,173)( 26,174)( 27,176)( 28,175)( 29,225)( 30,226)( 31,228)( 32,227)( 33,249)( 34,250)( 35,252)( 36,251)( 37,245)( 38,246)( 39,248)( 40,247)( 41,241)( 42,242)( 43,244)( 44,243)( 45,237)( 46,238)( 47,240)( 48,239)( 49,233)( 50,234)( 51,236)( 52,235)( 53,229)( 54,230)( 55,232)( 56,231)( 57,197)( 58,198)( 59,200)( 60,199)( 61,221)( 62,222)( 63,224)( 64,223)( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)( 72,215)( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)( 80,207)( 81,201)( 82,202)( 83,204)( 84,203)( 85,253)( 86,254)( 87,256)( 88,255)( 89,277)( 90,278)( 91,280)( 92,279)( 93,273)( 94,274)( 95,276)( 96,275)( 97,269)( 98,270)( 99,272)(100,271)(101,265)(102,266)(103,268)(104,267)(105,261)(106,262)(107,264)(108,263)(109,257)(110,258)(111,260)(112,259)(113,309)(114,310)(115,312)(116,311)(117,333)(118,334)(119,336)(120,335)(121,329)(122,330)(123,332)(124,331)(125,325)(126,326)(127,328)(128,327)(129,321)(130,322)(131,324)(132,323)(133,317)(134,318)(135,320)(136,319)(137,313)(138,314)(139,316)(140,315)(141,281)(142,282)(143,284)(144,283)(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)(151,304)(152,303)(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)(159,296)(160,295)(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)(167,288)(168,287);
s2 := 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)(169,201)(170,204)(171,203)(172,202)(173,197)(174,200)(175,199)(176,198)(177,221)(178,224)(179,223)(180,222)(181,217)(182,220)(183,219)(184,218)(185,213)(186,216)(187,215)(188,214)(189,209)(190,212)(191,211)(192,210)(193,205)(194,208)(195,207)(196,206)(225,229)(226,232)(227,231)(228,230)(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)(239,247)(240,246)(242,244)(253,285)(254,288)(255,287)(256,286)(257,281)(258,284)(259,283)(260,282)(261,305)(262,308)(263,307)(264,306)(265,301)(266,304)(267,303)(268,302)(269,297)(270,300)(271,299)(272,298)(273,293)(274,296)(275,295)(276,294)(277,289)(278,292)(279,291)(280,290)(309,313)(310,316)(311,315)(312,314)(317,333)(318,336)(319,335)(320,334)(321,329)(322,332)(323,331)(324,330)(326,328);
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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