Polytope of Type {18,42}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,42}*1512a
if this polytope has a name.
Group : SmallGroup(1512,485)
Rank : 3
Schlafli Type : {18,42}
Number of vertices, edges, etc : 18, 378, 42
Order of s0s1s2 : 126
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {18,14}*504, {6,42}*504a
   7-fold quotients : {18,6}*216b
   9-fold quotients : {6,14}*168
   14-fold quotients : {9,6}*108
   21-fold quotients : {18,2}*72, {6,6}*72c
   27-fold quotients : {2,14}*56
   42-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {2,7}*28
   63-fold quotients : {6,2}*24
   126-fold quotients : {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):
   None.

Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 22, 43)( 23, 45)( 24, 44)( 25, 46)( 26, 48)( 27, 47)( 28, 49)( 29, 51)( 30, 50)( 31, 52)( 32, 54)( 33, 53)( 34, 55)( 35, 57)( 36, 56)( 37, 58)( 38, 60)( 39, 59)( 40, 61)( 41, 63)( 42, 62)( 64,128)( 65,127)( 66,129)( 67,131)( 68,130)( 69,132)( 70,134)( 71,133)( 72,135)( 73,137)( 74,136)( 75,138)( 76,140)( 77,139)( 78,141)( 79,143)( 80,142)( 81,144)( 82,146)( 83,145)( 84,147)( 85,170)( 86,169)( 87,171)( 88,173)( 89,172)( 90,174)( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)( 96,180)( 97,182)( 98,181)( 99,183)(100,185)(101,184)(102,186)(103,188)(104,187)(105,189)(106,149)(107,148)(108,150)(109,152)(110,151)(111,153)(112,155)(113,154)(114,156)(115,158)(116,157)(117,159)(118,161)(119,160)(120,162)(121,164)(122,163)(123,165)(124,167)(125,166)(126,168);;
s1 := (  1, 85)(  2, 87)(  3, 86)(  4,103)(  5,105)(  6,104)(  7,100)(  8,102)(  9,101)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)( 16, 91)( 17, 93)( 18, 92)( 19, 88)( 20, 90)( 21, 89)( 22, 64)( 23, 66)( 24, 65)( 25, 82)( 26, 84)( 27, 83)( 28, 79)( 29, 81)( 30, 80)( 31, 76)( 32, 78)( 33, 77)( 34, 73)( 35, 75)( 36, 74)( 37, 70)( 38, 72)( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43,106)( 44,108)( 45,107)( 46,124)( 47,126)( 48,125)( 49,121)( 50,123)( 51,122)( 52,118)( 53,120)( 54,119)( 55,115)( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61,109)( 62,111)( 63,110)(127,149)(128,148)(129,150)(130,167)(131,166)(132,168)(133,164)(134,163)(135,165)(136,161)(137,160)(138,162)(139,158)(140,157)(141,159)(142,155)(143,154)(144,156)(145,152)(146,151)(147,153)(169,170)(172,188)(173,187)(174,189)(175,185)(176,184)(177,186)(178,182)(179,181)(180,183);;
s2 := (  1,  4)(  2,  5)(  3,  6)(  7, 19)(  8, 20)(  9, 21)( 10, 16)( 11, 17)( 12, 18)( 22, 46)( 23, 47)( 24, 48)( 25, 43)( 26, 44)( 27, 45)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 52)( 38, 53)( 39, 54)( 40, 49)( 41, 50)( 42, 51)( 64, 67)( 65, 68)( 66, 69)( 70, 82)( 71, 83)( 72, 84)( 73, 79)( 74, 80)( 75, 81)( 85,109)( 86,110)( 87,111)( 88,106)( 89,107)( 90,108)( 91,124)( 92,125)( 93,126)( 94,121)( 95,122)( 96,123)( 97,118)( 98,119)( 99,120)(100,115)(101,116)(102,117)(103,112)(104,113)(105,114)(127,130)(128,131)(129,132)(133,145)(134,146)(135,147)(136,142)(137,143)(138,144)(148,172)(149,173)(150,174)(151,169)(152,170)(153,171)(154,187)(155,188)(156,189)(157,184)(158,185)(159,186)(160,181)(161,182)(162,183)(163,178)(164,179)(165,180)(166,175)(167,176)(168,177);;
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*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(189)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 22, 43)( 23, 45)( 24, 44)( 25, 46)( 26, 48)( 27, 47)( 28, 49)( 29, 51)( 30, 50)( 31, 52)( 32, 54)( 33, 53)( 34, 55)( 35, 57)( 36, 56)( 37, 58)( 38, 60)( 39, 59)( 40, 61)( 41, 63)( 42, 62)( 64,128)( 65,127)( 66,129)( 67,131)( 68,130)( 69,132)( 70,134)( 71,133)( 72,135)( 73,137)( 74,136)( 75,138)( 76,140)( 77,139)( 78,141)( 79,143)( 80,142)( 81,144)( 82,146)( 83,145)( 84,147)( 85,170)( 86,169)( 87,171)( 88,173)( 89,172)( 90,174)( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)( 96,180)( 97,182)( 98,181)( 99,183)(100,185)(101,184)(102,186)(103,188)(104,187)(105,189)(106,149)(107,148)(108,150)(109,152)(110,151)(111,153)(112,155)(113,154)(114,156)(115,158)(116,157)(117,159)(118,161)(119,160)(120,162)(121,164)(122,163)(123,165)(124,167)(125,166)(126,168);
s1 := Sym(189)!(  1, 85)(  2, 87)(  3, 86)(  4,103)(  5,105)(  6,104)(  7,100)(  8,102)(  9,101)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)( 16, 91)( 17, 93)( 18, 92)( 19, 88)( 20, 90)( 21, 89)( 22, 64)( 23, 66)( 24, 65)( 25, 82)( 26, 84)( 27, 83)( 28, 79)( 29, 81)( 30, 80)( 31, 76)( 32, 78)( 33, 77)( 34, 73)( 35, 75)( 36, 74)( 37, 70)( 38, 72)( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43,106)( 44,108)( 45,107)( 46,124)( 47,126)( 48,125)( 49,121)( 50,123)( 51,122)( 52,118)( 53,120)( 54,119)( 55,115)( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61,109)( 62,111)( 63,110)(127,149)(128,148)(129,150)(130,167)(131,166)(132,168)(133,164)(134,163)(135,165)(136,161)(137,160)(138,162)(139,158)(140,157)(141,159)(142,155)(143,154)(144,156)(145,152)(146,151)(147,153)(169,170)(172,188)(173,187)(174,189)(175,185)(176,184)(177,186)(178,182)(179,181)(180,183);
s2 := Sym(189)!(  1,  4)(  2,  5)(  3,  6)(  7, 19)(  8, 20)(  9, 21)( 10, 16)( 11, 17)( 12, 18)( 22, 46)( 23, 47)( 24, 48)( 25, 43)( 26, 44)( 27, 45)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 52)( 38, 53)( 39, 54)( 40, 49)( 41, 50)( 42, 51)( 64, 67)( 65, 68)( 66, 69)( 70, 82)( 71, 83)( 72, 84)( 73, 79)( 74, 80)( 75, 81)( 85,109)( 86,110)( 87,111)( 88,106)( 89,107)( 90,108)( 91,124)( 92,125)( 93,126)( 94,121)( 95,122)( 96,123)( 97,118)( 98,119)( 99,120)(100,115)(101,116)(102,117)(103,112)(104,113)(105,114)(127,130)(128,131)(129,132)(133,145)(134,146)(135,147)(136,142)(137,143)(138,144)(148,172)(149,173)(150,174)(151,169)(152,170)(153,171)(154,187)(155,188)(156,189)(157,184)(158,185)(159,186)(160,181)(161,182)(162,183)(163,178)(164,179)(165,180)(166,175)(167,176)(168,177);
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*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*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 >;
References : None.
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