Polytope of Type {9,4}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4}*1296a
if this polytope has a name.
Group : SmallGroup(1296,3490)
Rank : 3
Schlafli Type : {9,4}
Number of vertices, edges, etc : 162, 324, 72
Order of s0s1s2 : 6
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {3,4}*48
54-fold quotients : {3,4}*24
108-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
36 facets:
36 of {9}*18
81 vertex figures:
81 of {4}*8
P/N, where N=<s1*s2*s1*s2> of order 2.
36 facets:
36 of {9}*18
84 vertex figures:
6 of {2}*4
78 of {4}*8
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
24 facets:
24 of {9}*18
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1> of order 3.
36 facets:
18 of {3}*6
18 of {9}*18
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.
24 facets:
24 of {9}*18
54 vertex figures:
54 of {4}*8
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
18 facets:
18 of {9}*18
45 vertex figures:
9 of {2}*4
36 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
12 facets:
12 of {9}*18
30 vertex figures:
24 of {4}*8
6 of {2}*4
P/N, where N=<s1*s2*s1*s2, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2> of order 6.
12 facets:
12 of {9}*18
30 vertex figures:
6 of {2}*4
24 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
12 facets:
12 of {9}*18
27 vertex figures:
27 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
18 facets:
9 of {3}*6
9 of {9}*18
27 vertex figures:
27 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 6.
12 facets:
12 of {9}*18
27 vertex figures:
27 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2> of order 9.
12 facets:
6 of {3}*6
6 of {9}*18
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 9.
16 facets:
4 of {9}*18
12 of {3}*6
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.
8 facets:
8 of {9}*18
18 vertex figures:
18 of {4}*8
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.
4 facets:
4 of {9}*18
12 vertex figures:
6 of {2}*4
6 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 18.
6 facets:
3 of {3}*6
3 of {9}*18
9 vertex figures:
9 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 18.
4 facets:
4 of {9}*18
9 vertex figures:
9 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 18.
8 facets:
2 of {9}*18
6 of {3}*6
9 vertex figures:
9 of {4}*8
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4,10)( 5,11)( 6,12)( 8, 9);;
s1 := ( 1, 2)( 4, 5)( 7,11)( 8,10)( 9,12);;
s2 := ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(12)!( 2, 3)( 4,10)( 5,11)( 6,12)( 8, 9);
s1 := Sym(12)!( 1, 2)( 4, 5)( 7,11)( 8,10)( 9,12);
s2 := Sym(12)!( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12);
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope
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