Polytope of Type {3,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,4}*192a
Also Known As : {{3,4}3,{4,4}4}. if this polytope has another name.
Group : SmallGroup(192,955)
Rank : 4
Schlafli Type : {3,4,4}
Number of vertices, edges, etc : 3, 12, 16, 8
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,4,4,2} of size 384
{3,4,4,3} of size 576
{3,4,4,4} of size 768
{3,4,4,6} of size 1152
{3,4,4,6} of size 1152
{3,4,4,9} of size 1728
{3,4,4,10} of size 1920
Vertex Figure Of :
{2,3,4,4} of size 384
{4,3,4,4} of size 768
{6,3,4,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
4-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,4,4}*384a, {3,4,4}*384b, {6,4,4}*384b, {6,4,4}*384c
3-fold covers : {9,4,4}*576a
4-fold covers : {3,4,8}*768a, {3,4,8}*768b, {3,8,4}*768a, {3,8,4}*768b, {3,4,4}*768a, {6,4,4}*768b, {12,4,4}*768c, {12,4,4}*768d, {3,4,4}*768b, {6,4,4}*768c, {6,4,4}*768d, {3,4,4}*768c, {3,8,4}*768e, {3,8,4}*768f, {6,4,4}*768f
5-fold covers : {15,4,4}*960a
6-fold covers : {9,4,4}*1152a, {9,4,4}*1152b, {18,4,4}*1152b, {18,4,4}*1152c, {3,4,12}*1152, {3,12,4}*1152a, {6,12,4}*1152d
7-fold covers : {21,4,4}*1344a
9-fold covers : {27,4,4}*1728a
10-fold covers : {15,4,4}*1920a, {3,4,20}*1920, {6,20,4}*1920b, {15,4,4}*1920b, {30,4,4}*1920b, {30,4,4}*1920c
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2,10)( 3,11)( 4,12);;
s1 := ( 5, 9)( 6,10)( 7,11)( 8,12);;
s2 := ( 1, 3)( 2, 4)( 9,11)(10,12);;
s3 := ( 3, 4)( 7, 8)(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,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s2*s1*s0*s2*s1*s0*s2*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(12)!( 1, 9)( 2,10)( 3,11)( 4,12);
s1 := Sym(12)!( 5, 9)( 6,10)( 7,11)( 8,12);
s2 := Sym(12)!( 1, 3)( 2, 4)( 9,11)(10,12);
s3 := Sym(12)!( 3, 4)( 7, 8)(11,12);
poly := sub<Sym(12)|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, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s2*s1*s0*s2*s1*s0*s2*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
References : None.
to this polytope