Polytope of Type {2,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,42}*168
if this polytope has a name.
Group : SmallGroup(168,56)
Rank : 3
Schlafli Type : {2,42}
Number of vertices, edges, etc : 2, 42, 42
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,42,2} of size 336
   {2,42,4} of size 672
   {2,42,4} of size 672
   {2,42,4} of size 672
   {2,42,6} of size 1008
   {2,42,6} of size 1008
   {2,42,6} of size 1008
   {2,42,8} of size 1344
   {2,42,6} of size 1344
   {2,42,4} of size 1344
   {2,42,6} of size 1512
   {2,42,10} of size 1680
Vertex Figure Of :
   {2,2,42} of size 336
   {3,2,42} of size 504
   {4,2,42} of size 672
   {5,2,42} of size 840
   {6,2,42} of size 1008
   {7,2,42} of size 1176
   {8,2,42} of size 1344
   {9,2,42} of size 1512
   {10,2,42} of size 1680
   {11,2,42} of size 1848
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,21}*84
   3-fold quotients : {2,14}*56
   6-fold quotients : {2,7}*28
   7-fold quotients : {2,6}*24
   14-fold quotients : {2,3}*12
   21-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,84}*336, {4,42}*336a
   3-fold covers : {2,126}*504, {6,42}*504b, {6,42}*504c
   4-fold covers : {4,84}*672a, {2,168}*672, {8,42}*672, {4,42}*672
   5-fold covers : {10,42}*840, {2,210}*840
   6-fold covers : {2,252}*1008, {4,126}*1008a, {12,42}*1008b, {6,84}*1008b, {6,84}*1008c, {12,42}*1008c
   7-fold covers : {2,294}*1176, {14,42}*1176b, {14,42}*1176c
   8-fold covers : {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b, {2,336}*1344, {16,42}*1344, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
   9-fold covers : {2,378}*1512, {6,126}*1512a, {6,126}*1512b, {18,42}*1512b, {6,42}*1512b, {6,42}*1512c, {6,42}*1512d
   10-fold covers : {20,42}*1680a, {10,84}*1680, {2,420}*1680, {4,210}*1680a
   11-fold covers : {22,42}*1848, {2,462}*1848
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)
(25,28)(26,27)(29,30)(31,34)(32,33)(35,36)(37,40)(38,39)(41,44)(42,43);;
s2 := ( 3,19)( 4,13)( 5,11)( 6,21)( 7, 9)( 8,31)(10,15)(12,25)(14,23)(16,33)
(17,20)(18,41)(22,27)(24,37)(26,35)(28,43)(29,32)(30,42)(34,39)(36,38)
(40,44);;
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*s1*s0*s1, s0*s2*s0*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*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(44)!(1,2);
s1 := Sym(44)!( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,21)
(23,24)(25,28)(26,27)(29,30)(31,34)(32,33)(35,36)(37,40)(38,39)(41,44)(42,43);
s2 := Sym(44)!( 3,19)( 4,13)( 5,11)( 6,21)( 7, 9)( 8,31)(10,15)(12,25)(14,23)
(16,33)(17,20)(18,41)(22,27)(24,37)(26,35)(28,43)(29,32)(30,42)(34,39)(36,38)
(40,44);
poly := sub<Sym(44)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

to this polytope