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Polytope of Type {6,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,42}*504c
if this polytope has a name.
Group : SmallGroup(504,192)
Rank : 3
Schlafli Type : {6,42}
Number of vertices, edges, etc : 6, 126, 42
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,42,2} of size 1008
Vertex Figure Of :
   {2,6,42} of size 1008
   {3,6,42} of size 1512
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,21}*252
   3-fold quotients : {2,42}*168
   6-fold quotients : {2,21}*84
   7-fold quotients : {6,6}*72b
   9-fold quotients : {2,14}*56
   14-fold quotients : {6,3}*36
   18-fold quotients : {2,7}*28
   21-fold quotients : {2,6}*24
   42-fold quotients : {2,3}*12
   63-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,84}*1008c, {12,42}*1008c
   3-fold covers : {6,126}*1512b, {6,42}*1512b, {6,42}*1512d
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)( 85,106)( 86,107)( 87,108)
( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)( 95,116)
( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)(103,124)
(104,125)(105,126);;
s1 := (  1, 22)(  2, 28)(  3, 27)(  4, 26)(  5, 25)(  6, 24)(  7, 23)(  8, 36)
(  9, 42)( 10, 41)( 11, 40)( 12, 39)( 13, 38)( 14, 37)( 15, 29)( 16, 35)
( 17, 34)( 18, 33)( 19, 32)( 20, 31)( 21, 30)( 44, 49)( 45, 48)( 46, 47)
( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 64, 85)
( 65, 91)( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)( 71, 99)( 72,105)
( 73,104)( 74,103)( 75,102)( 76,101)( 77,100)( 78, 92)( 79, 98)( 80, 97)
( 81, 96)( 82, 95)( 83, 94)( 84, 93)(107,112)(108,111)(109,110)(113,120)
(114,126)(115,125)(116,124)(117,123)(118,122)(119,121);;
s2 := (  1, 72)(  2, 71)(  3, 77)(  4, 76)(  5, 75)(  6, 74)(  7, 73)(  8, 65)
(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 79)( 16, 78)
( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 80)( 22,114)( 23,113)( 24,119)
( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)( 32,111)
( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)( 40,124)
( 41,123)( 42,122)( 43, 93)( 44, 92)( 45, 98)( 46, 97)( 47, 96)( 48, 95)
( 49, 94)( 50, 86)( 51, 85)( 52, 91)( 53, 90)( 54, 89)( 55, 88)( 56, 87)
( 57,100)( 58, 99)( 59,105)( 60,104)( 61,103)( 62,102)( 63,101);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*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*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(126)!( 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)( 85,106)( 86,107)
( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)
( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)
(103,124)(104,125)(105,126);
s1 := Sym(126)!(  1, 22)(  2, 28)(  3, 27)(  4, 26)(  5, 25)(  6, 24)(  7, 23)
(  8, 36)(  9, 42)( 10, 41)( 11, 40)( 12, 39)( 13, 38)( 14, 37)( 15, 29)
( 16, 35)( 17, 34)( 18, 33)( 19, 32)( 20, 31)( 21, 30)( 44, 49)( 45, 48)
( 46, 47)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)
( 64, 85)( 65, 91)( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)( 71, 99)
( 72,105)( 73,104)( 74,103)( 75,102)( 76,101)( 77,100)( 78, 92)( 79, 98)
( 80, 97)( 81, 96)( 82, 95)( 83, 94)( 84, 93)(107,112)(108,111)(109,110)
(113,120)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121);
s2 := Sym(126)!(  1, 72)(  2, 71)(  3, 77)(  4, 76)(  5, 75)(  6, 74)(  7, 73)
(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 79)
( 16, 78)( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 80)( 22,114)( 23,113)
( 24,119)( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)
( 32,111)( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)
( 40,124)( 41,123)( 42,122)( 43, 93)( 44, 92)( 45, 98)( 46, 97)( 47, 96)
( 48, 95)( 49, 94)( 50, 86)( 51, 85)( 52, 91)( 53, 90)( 54, 89)( 55, 88)
( 56, 87)( 57,100)( 58, 99)( 59,105)( 60,104)( 61,103)( 62,102)( 63,101);
poly := sub<Sym(126)|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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*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*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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