Polytope of Type {6,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,14}*1008a
Also Known As : {{6,6|2},{6,14|2}}. if this polytope has another name.
Group : SmallGroup(1008,922)
Rank : 4
Schlafli Type : {6,6,14}
Number of vertices, edges, etc : 6, 18, 42, 14
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 : {2,6,14}*336, {6,2,14}*336
   6-fold quotients : {3,2,14}*168, {6,2,7}*168
   7-fold quotients : {6,6,2}*144a
   9-fold quotients : {2,2,14}*112
   12-fold quotients : {3,2,7}*84
   18-fold quotients : {2,2,7}*56
   21-fold quotients : {2,6,2}*48, {6,2,2}*48
   42-fold quotients : {2,3,2}*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,  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);;
s2 := (  1, 22)(  2, 28)(  3, 27)(  4, 26)(  5, 25)(  6, 24)(  7, 23)(  8, 29)
(  9, 35)( 10, 34)( 11, 33)( 12, 32)( 13, 31)( 14, 30)( 15, 36)( 16, 42)
( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 44, 49)( 45, 48)( 46, 47)
( 51, 56)( 52, 55)( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 64, 85)( 65, 91)
( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)( 71, 92)( 72, 98)( 73, 97)
( 74, 96)( 75, 95)( 76, 94)( 77, 93)( 78, 99)( 79,105)( 80,104)( 81,103)
( 82,102)( 83,101)( 84,100)(107,112)(108,111)(109,110)(114,119)(115,118)
(116,117)(121,126)(122,125)(123,124);;
s3 := (  1, 65)(  2, 64)(  3, 70)(  4, 69)(  5, 68)(  6, 67)(  7, 66)(  8, 72)
(  9, 71)( 10, 77)( 11, 76)( 12, 75)( 13, 74)( 14, 73)( 15, 79)( 16, 78)
( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 80)( 22, 86)( 23, 85)( 24, 91)
( 25, 90)( 26, 89)( 27, 88)( 28, 87)( 29, 93)( 30, 92)( 31, 98)( 32, 97)
( 33, 96)( 34, 95)( 35, 94)( 36,100)( 37, 99)( 38,105)( 39,104)( 40,103)
( 41,102)( 42,101)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,108)( 50,114)( 51,113)( 52,119)( 53,118)( 54,117)( 55,116)( 56,115)
( 57,121)( 58,120)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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,  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);
s2 := Sym(126)!(  1, 22)(  2, 28)(  3, 27)(  4, 26)(  5, 25)(  6, 24)(  7, 23)
(  8, 29)(  9, 35)( 10, 34)( 11, 33)( 12, 32)( 13, 31)( 14, 30)( 15, 36)
( 16, 42)( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 44, 49)( 45, 48)
( 46, 47)( 51, 56)( 52, 55)( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 64, 85)
( 65, 91)( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)( 71, 92)( 72, 98)
( 73, 97)( 74, 96)( 75, 95)( 76, 94)( 77, 93)( 78, 99)( 79,105)( 80,104)
( 81,103)( 82,102)( 83,101)( 84,100)(107,112)(108,111)(109,110)(114,119)
(115,118)(116,117)(121,126)(122,125)(123,124);
s3 := Sym(126)!(  1, 65)(  2, 64)(  3, 70)(  4, 69)(  5, 68)(  6, 67)(  7, 66)
(  8, 72)(  9, 71)( 10, 77)( 11, 76)( 12, 75)( 13, 74)( 14, 73)( 15, 79)
( 16, 78)( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 80)( 22, 86)( 23, 85)
( 24, 91)( 25, 90)( 26, 89)( 27, 88)( 28, 87)( 29, 93)( 30, 92)( 31, 98)
( 32, 97)( 33, 96)( 34, 95)( 35, 94)( 36,100)( 37, 99)( 38,105)( 39,104)
( 40,103)( 41,102)( 42,101)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,108)( 50,114)( 51,113)( 52,119)( 53,118)( 54,117)( 55,116)
( 56,115)( 57,121)( 58,120)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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