Polytope of Type {14,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,6}*1008a
Also Known As : {{14,6|2},{6,6|2}}. if this polytope has another name.
Group : SmallGroup(1008,922)
Rank : 4
Schlafli Type : {14,6,6}
Number of vertices, edges, etc : 14, 42, 18, 6
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   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 : {14,2,6}*336, {14,6,2}*336
   6-fold quotients : {7,2,6}*168, {14,2,3}*168
   7-fold quotients : {2,6,6}*144a
   9-fold quotients : {14,2,2}*112
   12-fold quotients : {7,2,3}*84
   18-fold quotients : {7,2,2}*56
   21-fold quotients : {2,2,6}*48, {2,6,2}*48
   42-fold quotients : {2,2,3}*24, {2,3,2}*24
   63-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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