Polytope of Type {6,2,7}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,7}*168
if this polytope has a name.
Group : SmallGroup(168,50)
Rank : 4
Schlafli Type : {6,2,7}
Number of vertices, edges, etc : 6, 6, 7, 7
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,2,7,2} of size 336
Vertex Figure Of :
   {2,6,2,7} of size 336
   {3,6,2,7} of size 504
   {4,6,2,7} of size 672
   {3,6,2,7} of size 672
   {4,6,2,7} of size 672
   {4,6,2,7} of size 672
   {4,6,2,7} of size 1008
   {6,6,2,7} of size 1008
   {6,6,2,7} of size 1008
   {6,6,2,7} of size 1008
   {8,6,2,7} of size 1344
   {4,6,2,7} of size 1344
   {6,6,2,7} of size 1344
   {9,6,2,7} of size 1512
   {3,6,2,7} of size 1512
   {6,6,2,7} of size 1512
   {4,6,2,7} of size 1680
   {5,6,2,7} of size 1680
   {6,6,2,7} of size 1680
   {5,6,2,7} of size 1680
   {5,6,2,7} of size 1680
   {10,6,2,7} of size 1680
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,7}*84
   3-fold quotients : {2,2,7}*56
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,2,7}*336, {6,2,14}*336
   3-fold covers : {18,2,7}*504, {6,2,21}*504
   4-fold covers : {24,2,7}*672, {12,2,14}*672, {6,2,28}*672, {6,4,14}*672
   5-fold covers : {30,2,7}*840, {6,2,35}*840
   6-fold covers : {36,2,7}*1008, {18,2,14}*1008, {12,2,21}*1008, {6,6,14}*1008a, {6,6,14}*1008c, {6,2,42}*1008
   7-fold covers : {6,2,49}*1176, {6,14,7}*1176, {42,2,7}*1176
   8-fold covers : {48,2,7}*1344, {12,2,28}*1344, {12,4,14}*1344, {6,4,28}*1344, {24,2,14}*1344, {6,2,56}*1344, {6,8,14}*1344, {6,4,14}*1344
   9-fold covers : {54,2,7}*1512, {6,2,63}*1512, {18,2,21}*1512, {6,6,21}*1512a, {6,6,21}*1512b
   10-fold covers : {60,2,7}*1680, {12,2,35}*1680, {6,10,14}*1680, {30,2,14}*1680, {6,2,70}*1680
   11-fold covers : {66,2,7}*1848, {6,2,77}*1848
Permutation Representation (GAP) :

s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 8, 9)(10,11)(12,13);;
s3 := ( 7, 8)( 9,10)(11,12);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(13)!(3,4)(5,6);
s1 := Sym(13)!(1,5)(2,3)(4,6);
s2 := Sym(13)!( 8, 9)(10,11)(12,13);
s3 := Sym(13)!( 7, 8)( 9,10)(11,12);
poly := sub<Sym(13)|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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope