Polytope of Type {2,42,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,42,6}*1008b
if this polytope has a name.
Group : SmallGroup(1008,942)
Rank : 4
Schlafli Type : {2,42,6}
Number of vertices, edges, etc : 2, 42, 126, 6
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,14,6}*336, {2,42,2}*336
   6-fold quotients : {2,21,2}*168
   7-fold quotients : {2,6,6}*144a
   9-fold quotients : {2,14,2}*112
   18-fold quotients : {2,7,2}*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 := (1,2);;
s1 := (  4,  9)(  5,  8)(  6,  7)( 10, 17)( 11, 23)( 12, 22)( 13, 21)( 14, 20)
( 15, 19)( 16, 18)( 25, 30)( 26, 29)( 27, 28)( 31, 38)( 32, 44)( 33, 43)
( 34, 42)( 35, 41)( 36, 40)( 37, 39)( 46, 51)( 47, 50)( 48, 49)( 52, 59)
( 53, 65)( 54, 64)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 67, 72)( 68, 71)
( 69, 70)( 73, 80)( 74, 86)( 75, 85)( 76, 84)( 77, 83)( 78, 82)( 79, 81)
( 88, 93)( 89, 92)( 90, 91)( 94,101)( 95,107)( 96,106)( 97,105)( 98,104)
( 99,103)(100,102)(109,114)(110,113)(111,112)(115,122)(116,128)(117,127)
(118,126)(119,125)(120,124)(121,123);;
s2 := (  3, 11)(  4, 10)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)( 17, 18)
( 19, 23)( 20, 22)( 24, 53)( 25, 52)( 26, 58)( 27, 57)( 28, 56)( 29, 55)
( 30, 54)( 31, 46)( 32, 45)( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)
( 38, 60)( 39, 59)( 40, 65)( 41, 64)( 42, 63)( 43, 62)( 44, 61)( 66, 74)
( 67, 73)( 68, 79)( 69, 78)( 70, 77)( 71, 76)( 72, 75)( 80, 81)( 82, 86)
( 83, 85)( 87,116)( 88,115)( 89,121)( 90,120)( 91,119)( 92,118)( 93,117)
( 94,109)( 95,108)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,123)
(102,122)(103,128)(104,127)(105,126)(106,125)(107,124);;
s3 := (  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)( 22,106)( 23,107)( 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, 85)( 44, 86)( 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)( 64,127)( 65,128);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, 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*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(128)!(1,2);
s1 := Sym(128)!(  4,  9)(  5,  8)(  6,  7)( 10, 17)( 11, 23)( 12, 22)( 13, 21)
( 14, 20)( 15, 19)( 16, 18)( 25, 30)( 26, 29)( 27, 28)( 31, 38)( 32, 44)
( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 37, 39)( 46, 51)( 47, 50)( 48, 49)
( 52, 59)( 53, 65)( 54, 64)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 67, 72)
( 68, 71)( 69, 70)( 73, 80)( 74, 86)( 75, 85)( 76, 84)( 77, 83)( 78, 82)
( 79, 81)( 88, 93)( 89, 92)( 90, 91)( 94,101)( 95,107)( 96,106)( 97,105)
( 98,104)( 99,103)(100,102)(109,114)(110,113)(111,112)(115,122)(116,128)
(117,127)(118,126)(119,125)(120,124)(121,123);
s2 := Sym(128)!(  3, 11)(  4, 10)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)
( 17, 18)( 19, 23)( 20, 22)( 24, 53)( 25, 52)( 26, 58)( 27, 57)( 28, 56)
( 29, 55)( 30, 54)( 31, 46)( 32, 45)( 33, 51)( 34, 50)( 35, 49)( 36, 48)
( 37, 47)( 38, 60)( 39, 59)( 40, 65)( 41, 64)( 42, 63)( 43, 62)( 44, 61)
( 66, 74)( 67, 73)( 68, 79)( 69, 78)( 70, 77)( 71, 76)( 72, 75)( 80, 81)
( 82, 86)( 83, 85)( 87,116)( 88,115)( 89,121)( 90,120)( 91,119)( 92,118)
( 93,117)( 94,109)( 95,108)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)
(101,123)(102,122)(103,128)(104,127)(105,126)(106,125)(107,124);
s3 := Sym(128)!(  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)( 22,106)( 23,107)( 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, 85)( 44, 86)( 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)( 64,127)( 65,128);
poly := sub<Sym(128)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
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*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 >;

to this polytope