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}*1512d
if this polytope has a name.
Group : SmallGroup(1512,838)
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) :
   3-fold quotients : {6,42}*504a, {6,42}*504b, {6,42}*504c
   6-fold quotients : {6,21}*252
   7-fold quotients : {6,6}*216d
   9-fold quotients : {6,14}*168, {2,42}*168
   18-fold quotients : {2,21}*84
   21-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   27-fold quotients : {2,14}*56
   42-fold quotients : {3,6}*36, {6,3}*36
   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=<s1*s0*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*s1*s2> of order 2.
      63 facets:
         63 of {6}*12
      12 vertex figures:
         6 of {42}*84
         6 of {21}*42
   P/N, where N=<s0*s1*s0*s1> of order 3.
      84 facets:
         63 of {2}*4
         21 of {6}*12
      6 vertex figures:
         6 of {42}*84
   P/N, where N=<s0*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> of order 3.
      42 facets:
         42 of {6}*12
      12 vertex figures:
         3 of {42}*84
         9 of {14}*28

Permutation Representation (GAP) : 
s0 := ( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135)( 73,136)( 74,137)( 75,138)( 76,139)( 77,140)( 78,141)( 79,142)( 80,143)( 81,144)( 82,145)( 83,146)( 84,147)( 85,169)( 86,170)( 87,171)( 88,172)( 89,173)( 90,174)( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)( 97,181)( 98,182)( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);;
s1 := (  1, 85)(  2, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)(  8, 99)(  9,105)( 10,104)( 11,103)( 12,102)( 13,101)( 14,100)( 15, 92)( 16, 98)( 17, 97)( 18, 96)( 19, 95)( 20, 94)( 21, 93)( 22, 64)( 23, 70)( 24, 69)( 25, 68)( 26, 67)( 27, 66)( 28, 65)( 29, 78)( 30, 84)( 31, 83)( 32, 82)( 33, 81)( 34, 80)( 35, 79)( 36, 71)( 37, 77)( 38, 76)( 39, 75)( 40, 74)( 41, 73)( 42, 72)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)( 48,108)( 49,107)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)( 56,121)( 57,113)( 58,119)( 59,118)( 60,117)( 61,116)( 62,115)( 63,114)(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)(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184);;
s2 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 16)( 17, 21)( 18, 20)( 22, 51)( 23, 50)( 24, 56)( 25, 55)( 26, 54)( 27, 53)( 28, 52)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)( 64, 72)( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)( 81, 83)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)( 91,115)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,135)(128,134)(129,140)(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,177)(149,176)(150,182)(151,181)(152,180)(153,179)(154,178)(155,170)(156,169)(157,175)(158,174)(159,173)(160,172)(161,171)(162,184)(163,183)(164,189)(165,188)(166,187)(167,186)(168,185);;
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*s1*s2*s0*s1*s0*s1*s2*s1*s2*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(189)!( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135)( 73,136)( 74,137)( 75,138)( 76,139)( 77,140)( 78,141)( 79,142)( 80,143)( 81,144)( 82,145)( 83,146)( 84,147)( 85,169)( 86,170)( 87,171)( 88,172)( 89,173)( 90,174)( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)( 97,181)( 98,182)( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);
s1 := Sym(189)!(  1, 85)(  2, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)(  8, 99)(  9,105)( 10,104)( 11,103)( 12,102)( 13,101)( 14,100)( 15, 92)( 16, 98)( 17, 97)( 18, 96)( 19, 95)( 20, 94)( 21, 93)( 22, 64)( 23, 70)( 24, 69)( 25, 68)( 26, 67)( 27, 66)( 28, 65)( 29, 78)( 30, 84)( 31, 83)( 32, 82)( 33, 81)( 34, 80)( 35, 79)( 36, 71)( 37, 77)( 38, 76)( 39, 75)( 40, 74)( 41, 73)( 42, 72)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)( 48,108)( 49,107)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)( 56,121)( 57,113)( 58,119)( 59,118)( 60,117)( 61,116)( 62,115)( 63,114)(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)(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184);
s2 := Sym(189)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 16)( 17, 21)( 18, 20)( 22, 51)( 23, 50)( 24, 56)( 25, 55)( 26, 54)( 27, 53)( 28, 52)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)( 64, 72)( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)( 81, 83)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)( 91,115)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,135)(128,134)(129,140)(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,177)(149,176)(150,182)(151,181)(152,180)(153,179)(154,178)(155,170)(156,169)(157,175)(158,174)(159,173)(160,172)(161,171)(162,184)(163,183)(164,189)(165,188)(166,187)(167,186)(168,185);
poly := sub<Sym(189)|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*s1*s2*s0*s1*s0*s1*s2*s1*s2*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.
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