Polytope of Type {6,4}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*1296b
if this polytope has a name.
Group : SmallGroup(1296,3490)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 162, 324, 108
Order of s0s1s2 : 9
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
27-fold quotients : {6,4}*48b
54-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
56 facets:
4 of {3}*6
52 of {6}*12
81 vertex figures:
81 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 2.
54 facets:
54 of {6}*12
84 vertex figures:
78 of {4}*8
6 of {2}*4
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s1*s0*s1> of order 3.
38 facets:
3 of {2}*4
35 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
27 facets:
27 of {6}*12
45 vertex figures:
36 of {4}*8
9 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 6.
18 facets:
18 of {6}*12
30 vertex figures:
24 of {4}*8
6 of {2}*4
P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0> of order 6.
18 facets:
18 of {6}*12
30 vertex figures:
6 of {2}*4
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0> of order 6.
20 facets:
4 of {3}*6
16 of {6}*12
27 vertex figures:
27 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 6.
20 facets:
4 of {3}*6
16 of {6}*12
27 vertex figures:
27 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
20 facets:
4 of {3}*6
16 of {6}*12
27 vertex figures:
27 of {4}*8
P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
12 facets:
12 of {6}*12
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 9.
14 facets:
3 of {2}*4
11 of {6}*12
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2, s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
12 facets:
12 of {6}*12
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 9.
12 facets:
12 of {6}*12
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 12.
11 facets:
3 of {2}*4
8 of {6}*12
15 vertex figures:
12 of {4}*8
3 of {2}*4
P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 18.
6 facets:
6 of {6}*12
12 vertex figures:
6 of {2}*4
6 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 18.
8 facets:
4 of {3}*6
4 of {6}*12
9 vertex figures:
9 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 18.
8 facets:
4 of {3}*6
4 of {6}*12
9 vertex figures:
9 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 18.
8 facets:
4 of {3}*6
4 of {6}*12
9 vertex figures:
9 of {4}*8
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 27.
6 facets:
3 of {2}*4
3 of {6}*12
6 vertex figures:
6 of {4}*8
Permutation Representation (GAP) :
s0 := ( 4,10)( 5,12)( 6,11);;
s1 := ( 2, 3)( 5, 6)( 7,10)( 8,12)( 9,11);;
s2 := ( 1, 9)( 2, 7)( 3, 8)( 4,11)( 5,12)( 6,10);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(12)!( 4,10)( 5,12)( 6,11);
s1 := Sym(12)!( 2, 3)( 5, 6)( 7,10)( 8,12)( 9,11);
s2 := Sym(12)!( 1, 9)( 2, 7)( 3, 8)( 4,11)( 5,12)( 6,10);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 >;
References : None.
to this polytope
Twisty Puzzle