Polytope of Type {42,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,6}*1512d
if this polytope has a name.
Group : SmallGroup(1512,838)
Rank : 3
Schlafli Type : {42,6}
Number of vertices, edges, etc : 126, 378, 18
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {42,6}*504a, {42,6}*504b, {42,6}*504c
   6-fold quotients : {21,6}*252
   7-fold quotients : {6,6}*216d
   9-fold quotients : {14,6}*168, {42,2}*168
   18-fold quotients : {21,2}*84
   21-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   27-fold quotients : {14,2}*56
   42-fold quotients : {3,6}*36, {6,3}*36
   54-fold quotients : {7,2}*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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:
         6 of {21}*42
         6 of {42}*84
      63 vertex figures:
         63 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3.
      6 facets:
         6 of {42}*84
      84 vertex figures:
         21 of {6}*12
         63 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.
      12 facets:
         9 of {14}*28
         3 of {42}*84
      42 vertex figures:
         42 of {6}*12

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