Polytope of Type {6,6}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*576d
if this polytope has a name.
Group : SmallGroup(576,8654)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 48, 144, 48
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,6,2} of size 1152
Vertex Figure Of :
{2,6,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
16-fold quotients : {6,3}*36
48-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*1152h, {6,12}*1152i, {6,6}*1152k
3-fold covers : {6,18}*1728c, {6,6}*1728e
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 2.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
24 facets:
24 of {6}*12
28 vertex figures:
20 of {6}*12
8 of {3}*6
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
16 facets:
16 of {6}*12
20 vertex figures:
14 of {6}*12
6 of {2}*4
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 4.
12 facets:
12 of {6}*12
16 vertex figures:
8 of {3}*6
8 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 4.
12 facets:
12 of {6}*12
18 vertex figures:
6 of {6}*12
12 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
12 facets:
12 of {6}*12
14 vertex figures:
10 of {6}*12
4 of {3}*6
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 8.
6 facets:
6 of {6}*12
8 vertex figures:
4 of {6}*12
4 of {3}*6
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 8.
6 facets:
6 of {6}*12
10 vertex figures:
8 of {3}*6
2 of {6}*12
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 12.
4 facets:
4 of {6}*12
10 vertex figures:
6 of {2}*4
4 of {3}*6
Permutation Representation (GAP) :
s0 := ( 2, 3)( 7, 8)(10,12);;
s1 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12);;
s2 := ( 2, 3)( 5,11)( 6, 9)( 7,12)( 8,10);;
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*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(12)!( 2, 3)( 7, 8)(10,12);
s1 := Sym(12)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12);
s2 := Sym(12)!( 2, 3)( 5,11)( 6, 9)( 7,12)( 8,10);
poly := sub<Sym(12)|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*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1 >;
References : None.
to this polytope
Twisty Puzzle