Polytope of Type {4,6}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*1152
if this polytope has a name.
Group : SmallGroup(1152,157849)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 96, 288, 144
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
16-fold quotients : {4,6}*72
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
72 facets:
72 of {4}*8
48 vertex figures:
48 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
72 facets:
72 of {4}*8
48 vertex figures:
48 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
48 facets:
48 of {4}*8
32 vertex figures:
32 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
36 facets:
36 of {4}*8
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 8.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 8.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 16.
9 facets:
9 of {4}*8
6 vertex figures:
6 of {6}*12
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 16.
9 facets:
9 of {4}*8
6 vertex figures:
6 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 16.
9 facets:
9 of {4}*8
6 vertex figures:
6 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 16.
9 facets:
9 of {4}*8
6 vertex figures:
6 of {6}*12
Permutation Representation (GAP) :
s0 := ( 9,13)(10,14)(11,15)(12,16);;
s1 := ( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);;
s2 := ( 1, 5)( 2, 8)( 3, 7)( 4, 6)(10,12)(14,16);;
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*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(16)!( 9,13)(10,14)(11,15)(12,16);
s1 := Sym(16)!( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);
s2 := Sym(16)!( 1, 5)( 2, 8)( 3, 7)( 4, 6)(10,12)(14,16);
poly := sub<Sym(16)|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*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 >;
References : None.
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