Polytope of Type {6,42}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,42}*1512c
if this polytope has a name.
Group : SmallGroup(1512,561)
Rank : 3
Schlafli Type : {6,42}
Number of vertices, edges, etc : 18, 378, 126
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
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 : {6,42}*756
   3-fold quotients : {6,42}*504b
   7-fold quotients : {6,6}*216b
   9-fold quotients : {6,14}*168, {2,42}*168
   14-fold quotients : {6,6}*108
   18-fold quotients : {2,21}*84
   21-fold quotients : {6,6}*72a
   27-fold quotients : {2,14}*56
   54-fold quotients : {2,7}*28
   63-fold quotients : {2,6}*24, {6,2}*24
   126-fold quotients : {2,3}*12, {3,2}*12
   189-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> of order 3.
      70 facets:
         42 of {2}*4
         28 of {6}*12
      6 vertex figures:
         6 of {42}*84
   P/N, where N=<s0*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> of order 3.
      42 facets:
         42 of {6}*12
      10 vertex figures:
         4 of {42}*84
         6 of {14}*28

Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 64,127)( 65,129)( 66,128)( 67,130)( 68,132)( 69,131)( 70,133)( 71,135)( 72,134)( 73,136)( 74,138)( 75,137)( 76,139)( 77,141)( 78,140)( 79,142)( 80,144)( 81,143)( 82,145)( 83,147)( 84,146)( 85,148)( 86,150)( 87,149)( 88,151)( 89,153)( 90,152)( 91,154)( 92,156)( 93,155)( 94,157)( 95,159)( 96,158)( 97,160)( 98,162)( 99,161)(100,163)(101,165)(102,164)(103,166)(104,168)(105,167)(106,169)(107,171)(108,170)(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)(115,178)(116,180)(117,179)(118,181)(119,183)(120,182)(121,184)(122,186)(123,185)(124,187)(125,189)(126,188)(191,192)(194,195)(197,198)(200,201)(203,204)(206,207)(209,210)(212,213)(215,216)(218,219)(221,222)(224,225)(227,228)(230,231)(233,234)(236,237)(239,240)(242,243)(245,246)(248,249)(251,252)(253,316)(254,318)(255,317)(256,319)(257,321)(258,320)(259,322)(260,324)(261,323)(262,325)(263,327)(264,326)(265,328)(266,330)(267,329)(268,331)(269,333)(270,332)(271,334)(272,336)(273,335)(274,337)(275,339)(276,338)(277,340)(278,342)(279,341)(280,343)(281,345)(282,344)(283,346)(284,348)(285,347)(286,349)(287,351)(288,350)(289,352)(290,354)(291,353)(292,355)(293,357)(294,356)(295,358)(296,360)(297,359)(298,361)(299,363)(300,362)(301,364)(302,366)(303,365)(304,367)(305,369)(306,368)(307,370)(308,372)(309,371)(310,373)(311,375)(312,374)(313,376)(314,378)(315,377);;
s1 := (  1, 64)(  2, 65)(  3, 66)(  4, 82)(  5, 83)(  6, 84)(  7, 79)(  8, 80)(  9, 81)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)( 16, 70)( 17, 71)( 18, 72)( 19, 67)( 20, 68)( 21, 69)( 22,108)( 23,106)( 24,107)( 25,126)( 26,124)( 27,125)( 28,123)( 29,121)( 30,122)( 31,120)( 32,118)( 33,119)( 34,117)( 35,115)( 36,116)( 37,114)( 38,112)( 39,113)( 40,111)( 41,109)( 42,110)( 43, 86)( 44, 87)( 45, 85)( 46,104)( 47,105)( 48,103)( 49,101)( 50,102)( 51,100)( 52, 98)( 53, 99)( 54, 97)( 55, 95)( 56, 96)( 57, 94)( 58, 92)( 59, 93)( 60, 91)( 61, 89)( 62, 90)( 63, 88)(130,145)(131,146)(132,147)(133,142)(134,143)(135,144)(136,139)(137,140)(138,141)(148,171)(149,169)(150,170)(151,189)(152,187)(153,188)(154,186)(155,184)(156,185)(157,183)(158,181)(159,182)(160,180)(161,178)(162,179)(163,177)(164,175)(165,176)(166,174)(167,172)(168,173)(190,253)(191,254)(192,255)(193,271)(194,272)(195,273)(196,268)(197,269)(198,270)(199,265)(200,266)(201,267)(202,262)(203,263)(204,264)(205,259)(206,260)(207,261)(208,256)(209,257)(210,258)(211,297)(212,295)(213,296)(214,315)(215,313)(216,314)(217,312)(218,310)(219,311)(220,309)(221,307)(222,308)(223,306)(224,304)(225,305)(226,303)(227,301)(228,302)(229,300)(230,298)(231,299)(232,275)(233,276)(234,274)(235,293)(236,294)(237,292)(238,290)(239,291)(240,289)(241,287)(242,288)(243,286)(244,284)(245,285)(246,283)(247,281)(248,282)(249,280)(250,278)(251,279)(252,277)(319,334)(320,335)(321,336)(322,331)(323,332)(324,333)(325,328)(326,329)(327,330)(337,360)(338,358)(339,359)(340,378)(341,376)(342,377)(343,375)(344,373)(345,374)(346,372)(347,370)(348,371)(349,369)(350,367)(351,368)(352,366)(353,364)(354,365)(355,363)(356,361)(357,362);;
s2 := (  1,214)(  2,216)(  3,215)(  4,211)(  5,213)(  6,212)(  7,229)(  8,231)(  9,230)( 10,226)( 11,228)( 12,227)( 13,223)( 14,225)( 15,224)( 16,220)( 17,222)( 18,221)( 19,217)( 20,219)( 21,218)( 22,193)( 23,195)( 24,194)( 25,190)( 26,192)( 27,191)( 28,208)( 29,210)( 30,209)( 31,205)( 32,207)( 33,206)( 34,202)( 35,204)( 36,203)( 37,199)( 38,201)( 39,200)( 40,196)( 41,198)( 42,197)( 43,235)( 44,237)( 45,236)( 46,232)( 47,234)( 48,233)( 49,250)( 50,252)( 51,251)( 52,247)( 53,249)( 54,248)( 55,244)( 56,246)( 57,245)( 58,241)( 59,243)( 60,242)( 61,238)( 62,240)( 63,239)( 64,277)( 65,279)( 66,278)( 67,274)( 68,276)( 69,275)( 70,292)( 71,294)( 72,293)( 73,289)( 74,291)( 75,290)( 76,286)( 77,288)( 78,287)( 79,283)( 80,285)( 81,284)( 82,280)( 83,282)( 84,281)( 85,256)( 86,258)( 87,257)( 88,253)( 89,255)( 90,254)( 91,271)( 92,273)( 93,272)( 94,268)( 95,270)( 96,269)( 97,265)( 98,267)( 99,266)(100,262)(101,264)(102,263)(103,259)(104,261)(105,260)(106,298)(107,300)(108,299)(109,295)(110,297)(111,296)(112,313)(113,315)(114,314)(115,310)(116,312)(117,311)(118,307)(119,309)(120,308)(121,304)(122,306)(123,305)(124,301)(125,303)(126,302)(127,340)(128,342)(129,341)(130,337)(131,339)(132,338)(133,355)(134,357)(135,356)(136,352)(137,354)(138,353)(139,349)(140,351)(141,350)(142,346)(143,348)(144,347)(145,343)(146,345)(147,344)(148,319)(149,321)(150,320)(151,316)(152,318)(153,317)(154,334)(155,336)(156,335)(157,331)(158,333)(159,332)(160,328)(161,330)(162,329)(163,325)(164,327)(165,326)(166,322)(167,324)(168,323)(169,361)(170,363)(171,362)(172,358)(173,360)(174,359)(175,376)(176,378)(177,377)(178,373)(179,375)(180,374)(181,370)(182,372)(183,371)(184,367)(185,369)(186,368)(187,364)(188,366)(189,365);;
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*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(378)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 64,127)( 65,129)( 66,128)( 67,130)( 68,132)( 69,131)( 70,133)( 71,135)( 72,134)( 73,136)( 74,138)( 75,137)( 76,139)( 77,141)( 78,140)( 79,142)( 80,144)( 81,143)( 82,145)( 83,147)( 84,146)( 85,148)( 86,150)( 87,149)( 88,151)( 89,153)( 90,152)( 91,154)( 92,156)( 93,155)( 94,157)( 95,159)( 96,158)( 97,160)( 98,162)( 99,161)(100,163)(101,165)(102,164)(103,166)(104,168)(105,167)(106,169)(107,171)(108,170)(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)(115,178)(116,180)(117,179)(118,181)(119,183)(120,182)(121,184)(122,186)(123,185)(124,187)(125,189)(126,188)(191,192)(194,195)(197,198)(200,201)(203,204)(206,207)(209,210)(212,213)(215,216)(218,219)(221,222)(224,225)(227,228)(230,231)(233,234)(236,237)(239,240)(242,243)(245,246)(248,249)(251,252)(253,316)(254,318)(255,317)(256,319)(257,321)(258,320)(259,322)(260,324)(261,323)(262,325)(263,327)(264,326)(265,328)(266,330)(267,329)(268,331)(269,333)(270,332)(271,334)(272,336)(273,335)(274,337)(275,339)(276,338)(277,340)(278,342)(279,341)(280,343)(281,345)(282,344)(283,346)(284,348)(285,347)(286,349)(287,351)(288,350)(289,352)(290,354)(291,353)(292,355)(293,357)(294,356)(295,358)(296,360)(297,359)(298,361)(299,363)(300,362)(301,364)(302,366)(303,365)(304,367)(305,369)(306,368)(307,370)(308,372)(309,371)(310,373)(311,375)(312,374)(313,376)(314,378)(315,377);
s1 := Sym(378)!(  1, 64)(  2, 65)(  3, 66)(  4, 82)(  5, 83)(  6, 84)(  7, 79)(  8, 80)(  9, 81)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)( 16, 70)( 17, 71)( 18, 72)( 19, 67)( 20, 68)( 21, 69)( 22,108)( 23,106)( 24,107)( 25,126)( 26,124)( 27,125)( 28,123)( 29,121)( 30,122)( 31,120)( 32,118)( 33,119)( 34,117)( 35,115)( 36,116)( 37,114)( 38,112)( 39,113)( 40,111)( 41,109)( 42,110)( 43, 86)( 44, 87)( 45, 85)( 46,104)( 47,105)( 48,103)( 49,101)( 50,102)( 51,100)( 52, 98)( 53, 99)( 54, 97)( 55, 95)( 56, 96)( 57, 94)( 58, 92)( 59, 93)( 60, 91)( 61, 89)( 62, 90)( 63, 88)(130,145)(131,146)(132,147)(133,142)(134,143)(135,144)(136,139)(137,140)(138,141)(148,171)(149,169)(150,170)(151,189)(152,187)(153,188)(154,186)(155,184)(156,185)(157,183)(158,181)(159,182)(160,180)(161,178)(162,179)(163,177)(164,175)(165,176)(166,174)(167,172)(168,173)(190,253)(191,254)(192,255)(193,271)(194,272)(195,273)(196,268)(197,269)(198,270)(199,265)(200,266)(201,267)(202,262)(203,263)(204,264)(205,259)(206,260)(207,261)(208,256)(209,257)(210,258)(211,297)(212,295)(213,296)(214,315)(215,313)(216,314)(217,312)(218,310)(219,311)(220,309)(221,307)(222,308)(223,306)(224,304)(225,305)(226,303)(227,301)(228,302)(229,300)(230,298)(231,299)(232,275)(233,276)(234,274)(235,293)(236,294)(237,292)(238,290)(239,291)(240,289)(241,287)(242,288)(243,286)(244,284)(245,285)(246,283)(247,281)(248,282)(249,280)(250,278)(251,279)(252,277)(319,334)(320,335)(321,336)(322,331)(323,332)(324,333)(325,328)(326,329)(327,330)(337,360)(338,358)(339,359)(340,378)(341,376)(342,377)(343,375)(344,373)(345,374)(346,372)(347,370)(348,371)(349,369)(350,367)(351,368)(352,366)(353,364)(354,365)(355,363)(356,361)(357,362);
s2 := Sym(378)!(  1,214)(  2,216)(  3,215)(  4,211)(  5,213)(  6,212)(  7,229)(  8,231)(  9,230)( 10,226)( 11,228)( 12,227)( 13,223)( 14,225)( 15,224)( 16,220)( 17,222)( 18,221)( 19,217)( 20,219)( 21,218)( 22,193)( 23,195)( 24,194)( 25,190)( 26,192)( 27,191)( 28,208)( 29,210)( 30,209)( 31,205)( 32,207)( 33,206)( 34,202)( 35,204)( 36,203)( 37,199)( 38,201)( 39,200)( 40,196)( 41,198)( 42,197)( 43,235)( 44,237)( 45,236)( 46,232)( 47,234)( 48,233)( 49,250)( 50,252)( 51,251)( 52,247)( 53,249)( 54,248)( 55,244)( 56,246)( 57,245)( 58,241)( 59,243)( 60,242)( 61,238)( 62,240)( 63,239)( 64,277)( 65,279)( 66,278)( 67,274)( 68,276)( 69,275)( 70,292)( 71,294)( 72,293)( 73,289)( 74,291)( 75,290)( 76,286)( 77,288)( 78,287)( 79,283)( 80,285)( 81,284)( 82,280)( 83,282)( 84,281)( 85,256)( 86,258)( 87,257)( 88,253)( 89,255)( 90,254)( 91,271)( 92,273)( 93,272)( 94,268)( 95,270)( 96,269)( 97,265)( 98,267)( 99,266)(100,262)(101,264)(102,263)(103,259)(104,261)(105,260)(106,298)(107,300)(108,299)(109,295)(110,297)(111,296)(112,313)(113,315)(114,314)(115,310)(116,312)(117,311)(118,307)(119,309)(120,308)(121,304)(122,306)(123,305)(124,301)(125,303)(126,302)(127,340)(128,342)(129,341)(130,337)(131,339)(132,338)(133,355)(134,357)(135,356)(136,352)(137,354)(138,353)(139,349)(140,351)(141,350)(142,346)(143,348)(144,347)(145,343)(146,345)(147,344)(148,319)(149,321)(150,320)(151,316)(152,318)(153,317)(154,334)(155,336)(156,335)(157,331)(158,333)(159,332)(160,328)(161,330)(162,329)(163,325)(164,327)(165,326)(166,322)(167,324)(168,323)(169,361)(170,363)(171,362)(172,358)(173,360)(174,359)(175,376)(176,378)(177,377)(178,373)(179,375)(180,374)(181,370)(182,372)(183,371)(184,367)(185,369)(186,368)(187,364)(188,366)(189,365);
poly := sub<Sym(378)|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*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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 >;
References : None.
to this polytope

Twisty Puzzle