Polytope of Type {6,14,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,14,6}*1008
Also Known As : {{6,14|2},{14,6|2}}. if this polytope has another name.
Group : SmallGroup(1008,922)
Rank : 4
Schlafli Type : {6,14,6}
Number of vertices, edges, etc : 6, 42, 42, 6
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,14,6}*336, {6,14,2}*336
   7-fold quotients : {6,2,6}*144
   9-fold quotients : {2,14,2}*112
   14-fold quotients : {3,2,6}*72, {6,2,3}*72
   18-fold quotients : {2,7,2}*56
   21-fold quotients : {2,2,6}*48, {6,2,2}*48
   28-fold quotients : {3,2,3}*36
   42-fold quotients : {2,2,3}*24, {3,2,2}*24
   63-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)
(118,125)(119,126);;
s1 := (  1,  8)(  2, 14)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 16, 21)
( 17, 20)( 18, 19)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)( 27, 31)
( 28, 30)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 58, 63)( 59, 62)( 60, 61)( 64, 71)( 65, 77)
( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 79, 84)( 80, 83)( 81, 82)
( 85, 92)( 86, 98)( 87, 97)( 88, 96)( 89, 95)( 90, 94)( 91, 93)(100,105)
(101,104)(102,103)(106,113)(107,119)(108,118)(109,117)(110,116)(111,115)
(112,114)(121,126)(122,125)(123,124);;
s2 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 44)( 23, 43)( 24, 49)( 25, 48)( 26, 47)( 27, 46)( 28, 45)
( 29, 51)( 30, 50)( 31, 56)( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 58)
( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)( 64, 65)( 66, 70)
( 67, 69)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,107)
( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)
( 94,119)( 95,118)( 96,117)( 97,116)( 98,115)( 99,121)(100,120)(101,126)
(102,125)(103,124)(104,123)(105,122);;
s3 := (  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)(  8, 92)
(  9, 93)( 10, 94)( 11, 95)( 12, 96)( 13, 97)( 14, 98)( 15, 99)( 16,100)
( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22, 64)( 23, 65)( 24, 66)
( 25, 67)( 26, 68)( 27, 69)( 28, 70)( 29, 71)( 30, 72)( 31, 73)( 32, 74)
( 33, 75)( 34, 76)( 35, 77)( 36, 78)( 37, 79)( 38, 80)( 39, 81)( 40, 82)
( 41, 83)( 42, 84)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)( 48,111)
( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,118)( 56,119)
( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)!(  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)
( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)
( 95,102)( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)
(117,124)(118,125)(119,126);
s1 := Sym(126)!(  1,  8)(  2, 14)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)
( 16, 21)( 17, 20)( 18, 19)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)
( 27, 31)( 28, 30)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)
( 46, 54)( 47, 53)( 48, 52)( 49, 51)( 58, 63)( 59, 62)( 60, 61)( 64, 71)
( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 79, 84)( 80, 83)
( 81, 82)( 85, 92)( 86, 98)( 87, 97)( 88, 96)( 89, 95)( 90, 94)( 91, 93)
(100,105)(101,104)(102,103)(106,113)(107,119)(108,118)(109,117)(110,116)
(111,115)(112,114)(121,126)(122,125)(123,124);
s2 := Sym(126)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 44)( 23, 43)( 24, 49)( 25, 48)( 26, 47)( 27, 46)
( 28, 45)( 29, 51)( 30, 50)( 31, 56)( 32, 55)( 33, 54)( 34, 53)( 35, 52)
( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)( 64, 65)
( 66, 70)( 67, 69)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)
( 85,107)( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,114)
( 93,113)( 94,119)( 95,118)( 96,117)( 97,116)( 98,115)( 99,121)(100,120)
(101,126)(102,125)(103,124)(104,123)(105,122);
s3 := Sym(126)!(  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)
(  8, 92)(  9, 93)( 10, 94)( 11, 95)( 12, 96)( 13, 97)( 14, 98)( 15, 99)
( 16,100)( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22, 64)( 23, 65)
( 24, 66)( 25, 67)( 26, 68)( 27, 69)( 28, 70)( 29, 71)( 30, 72)( 31, 73)
( 32, 74)( 33, 75)( 34, 76)( 35, 77)( 36, 78)( 37, 79)( 38, 80)( 39, 81)
( 40, 82)( 41, 83)( 42, 84)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)
( 48,111)( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,118)
( 56,119)( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126);
poly := sub<Sym(126)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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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