Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*72
Also Known As : {4,4}_(3,0), {4,4|3}. if this polytope has another name.
Group : SmallGroup(72,40)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 9, 18, 9
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 3
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   {4,4,2} of size 144
   {4,4,3} of size 720
   {4,4,4} of size 720
   {4,4,4} of size 1152
   {4,4,4} of size 1152
   {4,4,6} of size 1152
   {4,4,6} of size 1296
   {4,4,3} of size 1440
   {4,4,4} of size 1440
   {4,4,4} of size 1440
   {4,4,4} of size 1440
   {4,4,6} of size 1440
   {4,4,6} of size 1440
Vertex Figure Of :
   {2,4,4} of size 144
   {3,4,4} of size 720
   {4,4,4} of size 720
   {4,4,4} of size 1152
   {4,4,4} of size 1152
   {6,4,4} of size 1152
   {6,4,4} of size 1296
   {3,4,4} of size 1440
   {4,4,4} of size 1440
   {4,4,4} of size 1440
   {4,4,4} of size 1440
   {6,4,4} of size 1440
   {6,4,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4}*144
   3-fold covers : {4,12}*216, {12,4}*216
   4-fold covers : {4,4}*288
   6-fold covers : {4,12}*432a, {12,4}*432a, {4,12}*432b, {12,4}*432b
   8-fold covers : {4,4}*576, {4,8}*576a, {8,4}*576a, {4,8}*576b, {8,4}*576b
   9-fold covers : {4,4}*648, {12,12}*648
   10-fold covers : {4,20}*720, {20,4}*720
   12-fold covers : {4,12}*864a, {12,4}*864a, {4,12}*864d, {12,4}*864c, {12,12}*864m
   14-fold covers : {4,28}*1008, {28,4}*1008
   16-fold covers : {4,8}*1152a, {8,4}*1152a, {8,8}*1152a, {8,8}*1152b, {8,8}*1152c, {8,8}*1152d, {4,16}*1152a, {16,4}*1152a, {4,16}*1152b, {16,4}*1152b, {4,8}*1152b, {8,4}*1152b, {4,4}*1152, {4,8}*1152c, {4,8}*1152d, {8,4}*1152c, {8,4}*1152d, {8,8}*1152e, {8,8}*1152f
   18-fold covers : {4,4}*1296, {4,36}*1296, {36,4}*1296, {4,12}*1296, {12,4}*1296, {12,12}*1296a, {12,12}*1296b, {12,12}*1296c, {12,12}*1296d, {12,12}*1296e, {12,12}*1296f, {12,12}*1296g, {12,12}*1296h
   20-fold covers : {4,20}*1440, {20,4}*1440
   22-fold covers : {4,44}*1584, {44,4}*1584
   24-fold covers : {4,12}*1728b, {12,4}*1728a, {4,24}*1728b, {8,12}*1728b, {12,8}*1728b, {24,4}*1728b, {4,24}*1728d, {8,12}*1728c, {12,8}*1728c, {24,4}*1728d, {4,24}*1728f, {24,4}*1728e, {8,12}*1728e, {12,8}*1728e, {4,24}*1728h, {24,4}*1728h, {8,12}*1728f, {12,8}*1728f, {4,12}*1728d, {12,4}*1728d, {4,12}*1728e, {12,4}*1728e, {12,12}*1728z
   25-fold covers : {4,4}*1800
   26-fold covers : {4,52}*1872, {52,4}*1872
   27-fold covers : {4,12}*1944a, {12,4}*1944a, {4,12}*1944b, {4,12}*1944c, {12,4}*1944b, {12,4}*1944c, {12,12}*1944a, {12,12}*1944b, {4,12}*1944d, {12,4}*1944d, {12,12}*1944c, {12,12}*1944d, {12,12}*1944e, {12,12}*1944f
Permutation Representation (GAP) :

s0 := (5,6);;
s1 := (1,2)(3,5)(4,6);;
s2 := (2,3);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(6)!(5,6);
s1 := Sym(6)!(1,2)(3,5)(4,6);
s2 := Sym(6)!(2,3);
poly := sub<Sym(6)|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, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope