Polytope of Type {4,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,42}*1344a
if this polytope has a name.
Group : SmallGroup(1344,6453)
Rank : 3
Schlafli Type : {4,42}
Number of vertices, edges, etc : 16, 336, 168
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {4,42}*336b
   7-fold quotients : {4,6}*192a
   8-fold quotients : {4,21}*168
   28-fold quotients : {4,6}*48c
   56-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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