Polytope of Type {6,5}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,5}*1920c
if this polytope has a name.
Group : SmallGroup(1920,240997)
Rank : 3
Schlafli Type : {6,5}
Number of vertices, edges, etc : 192, 480, 160
Order of s0s1s2 : 10
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,5}*960
16-fold quotients : {6,5}*120b
32-fold quotients : {3,5}*60
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*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 2.
80 facets:
80 of {6}*12
96 vertex figures:
96 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
80 facets:
80 of {6}*12
96 vertex figures:
96 of {5}*10
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 2.
80 facets:
80 of {6}*12
96 vertex figures:
96 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0> of order 2.
88 facets:
72 of {6}*12
16 of {3}*6
96 vertex figures:
96 of {5}*10
P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 3.
56 facets:
52 of {6}*12
4 of {2}*4
64 vertex figures:
64 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
40 facets:
40 of {6}*12
48 vertex figures:
48 of {5}*10
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
40 facets:
40 of {6}*12
48 vertex figures:
48 of {5}*10
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
40 facets:
40 of {6}*12
48 vertex figures:
48 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
40 facets:
40 of {6}*12
48 vertex figures:
48 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
48 facets:
32 of {6}*12
16 of {3}*6
48 vertex figures:
48 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
44 facets:
36 of {6}*12
8 of {3}*6
48 vertex figures:
48 of {5}*10
P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2> of order 4.
48 facets:
32 of {6}*12
16 of {3}*6
48 vertex figures:
48 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
44 facets:
36 of {6}*12
8 of {3}*6
48 vertex figures:
48 of {5}*10
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
40 facets:
40 of {6}*12
48 vertex figures:
48 of {5}*10
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 6.
28 facets:
26 of {6}*12
2 of {2}*4
32 vertex figures:
32 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 8.
20 facets:
20 of {6}*12
24 vertex figures:
24 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 8.
22 facets:
18 of {6}*12
4 of {3}*6
24 vertex figures:
24 of {5}*10
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 8.
20 facets:
20 of {6}*12
24 vertex figures:
24 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
28 facets:
12 of {6}*12
16 of {3}*6
24 vertex figures:
24 of {5}*10
P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2> of order 8.
26 facets:
14 of {6}*12
12 of {3}*6
24 vertex figures:
24 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
20 facets:
20 of {6}*12
24 vertex figures:
24 of {5}*10
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2> of order 8.
24 facets:
16 of {6}*12
8 of {3}*6
24 vertex figures:
24 of {5}*10
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
26 facets:
12 of {3}*6
14 of {6}*12
24 vertex figures:
24 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 12.
16 facets:
12 of {6}*12
4 of {2}*4
16 vertex figures:
16 of {5}*10
P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 16.
14 facets:
6 of {6}*12
8 of {3}*6
12 vertex figures:
12 of {5}*10
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 16.
16 facets:
4 of {6}*12
12 of {3}*6
12 vertex figures:
12 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 16.
10 facets:
10 of {6}*12
12 vertex figures:
12 of {5}*10
P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 24.
10 facets:
4 of {6}*12
4 of {3}*6
2 of {2}*4
8 vertex figures:
8 of {5}*10
Permutation Representation (GAP) :
s0 := ( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,10);;
s1 := ( 1, 3)( 2, 4)( 6, 9)( 8,10);;
s2 := ( 3, 6)( 4, 8)( 5, 9)( 7,10);;
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*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2,
s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(10)!( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,10);
s1 := Sym(10)!( 1, 3)( 2, 4)( 6, 9)( 8,10);
s2 := Sym(10)!( 3, 6)( 4, 8)( 5, 9)( 7,10);
poly := sub<Sym(10)|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*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2,
s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1 >;
References : None.
to this polytope
Twisty Puzzle