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}*1344
if this polytope has a name.
Group : SmallGroup(1344,11684)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 168, 336, 112
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
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) :
2-fold quotients : {6,4}*672a, {6,4}*672b, {6,4}*672c
4-fold quotients : {6,4}*336
168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
56 facets:
56 of {6}*12
88 vertex figures:
80 of {4}*8
8 of {2}*4
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 2.
56 facets:
56 of {6}*12
84 vertex figures:
84 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 2.
56 facets:
56 of {6}*12
84 vertex figures:
84 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
58 facets:
54 of {6}*12
4 of {3}*6
84 vertex figures:
84 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
40 facets:
36 of {6}*12
4 of {2}*4
56 vertex figures:
56 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
28 facets:
28 of {6}*12
48 vertex figures:
36 of {4}*8
12 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
28 facets:
28 of {6}*12
44 vertex figures:
40 of {4}*8
4 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
28 facets:
28 of {6}*12
44 vertex figures:
40 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
28 facets:
28 of {6}*12
44 vertex figures:
40 of {4}*8
4 of {2}*4
P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
30 facets:
26 of {6}*12
4 of {3}*6
44 vertex figures:
40 of {4}*8
4 of {2}*4
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
28 facets:
28 of {6}*12
44 vertex figures:
40 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 6.
22 facets:
4 of {3}*6
16 of {6}*12
2 of {2}*4
28 vertex figures:
28 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
20 facets:
18 of {6}*12
2 of {2}*4
28 vertex figures:
28 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
20 facets:
18 of {6}*12
2 of {2}*4
32 vertex figures:
24 of {4}*8
8 of {2}*4
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
20 facets:
18 of {6}*12
2 of {2}*4
28 vertex figures:
28 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 7.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 8.
15 facets:
13 of {6}*12
2 of {3}*6
22 vertex figures:
20 of {4}*8
2 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 12.
12 facets:
8 of {6}*12
4 of {2}*4
16 vertex figures:
12 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 14.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 14.
10 facets:
6 of {6}*12
4 of {3}*6
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0> of order 21.
8 facets:
4 of {6}*12
4 of {2}*4
8 vertex figures:
8 of {4}*8
Permutation Representation (GAP) :
s0 := (3,8)(4,7)(5,6);;
s1 := ( 1, 3)( 2, 6)( 4, 7)( 5, 8)( 9,10)(11,12);;
s2 := ( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,11)(10,12);;
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*s0*s1*s0*s1*s0*s1*s0*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,
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(12)!(3,8)(4,7)(5,6);
s1 := Sym(12)!( 1, 3)( 2, 6)( 4, 7)( 5, 8)( 9,10)(11,12);
s2 := Sym(12)!( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,11)(10,12);
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*s0*s1*s0*s1*s0*s1*s0*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,
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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