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}*960
if this polytope has a name.
Group : SmallGroup(960,10871)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 80, 240, 120
Order of s0s1s2 : 20
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,6,2} of size 1920
Vertex Figure Of :
{2,4,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6}*480
4-fold quotients : {4,6}*240a, {4,6}*240b, {4,6}*240c
8-fold quotients : {4,6}*120
60-fold quotients : {4,2}*16
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12}*1920a, {8,6}*1920a, {8,6}*1920b, {4,12}*1920b, {4,6}*1920
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
60 facets:
60 of {4}*8
40 vertex figures:
40 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
60 facets:
60 of {4}*8
40 vertex figures:
40 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 2.
60 facets:
60 of {4}*8
40 vertex figures:
40 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2> of order 2.
64 facets:
56 of {4}*8
8 of {2}*4
40 vertex figures:
40 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
60 facets:
60 of {4}*8
40 vertex figures:
40 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
60 facets:
60 of {4}*8
42 vertex figures:
4 of {3}*6
38 of {6}*12
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
40 facets:
40 of {4}*8
32 vertex figures:
24 of {6}*12
8 of {2}*4
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
30 facets:
30 of {4}*8
20 vertex figures:
20 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
30 facets:
30 of {4}*8
20 vertex figures:
20 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
30 facets:
30 of {4}*8
20 vertex figures:
20 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
34 facets:
8 of {2}*4
26 of {4}*8
20 vertex figures:
20 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
30 facets:
30 of {4}*8
21 vertex figures:
19 of {6}*12
2 of {3}*6
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
32 facets:
28 of {4}*8
4 of {2}*4
22 vertex figures:
4 of {3}*6
18 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 4.
30 facets:
30 of {4}*8
22 vertex figures:
4 of {3}*6
18 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
20 facets:
20 of {4}*8
16 vertex figures:
12 of {6}*12
4 of {2}*4
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 6.
20 facets:
20 of {4}*8
16 vertex figures:
12 of {6}*12
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
24 facets:
16 of {4}*8
8 of {2}*4
16 vertex figures:
12 of {6}*12
4 of {2}*4
P/N, where N=<s1*s2*s1*s2*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
20 facets:
20 of {4}*8
18 vertex figures:
4 of {3}*6
10 of {6}*12
4 of {2}*4
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
20 facets:
20 of {4}*8
16 vertex figures:
12 of {6}*12
4 of {2}*4
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 8.
17 facets:
4 of {2}*4
13 of {4}*8
10 vertex figures:
10 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2> of order 8.
17 facets:
4 of {2}*4
13 of {4}*8
11 vertex figures:
2 of {3}*6
9 of {6}*12
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 10.
16 facets:
8 of {4}*8
8 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 10.
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*s0*s1*s2*s1*s0*s1*s2*s1> of order 10.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 12.
10 facets:
10 of {4}*8
12 vertex figures:
4 of {6}*12
8 of {2}*4
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
10 facets:
10 of {4}*8
9 vertex figures:
5 of {6}*12
2 of {3}*6
2 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 20.
6 facets:
6 of {4}*8
4 vertex figures:
4 of {6}*12
Permutation Representation (GAP) :
s0 := (2,3)(7,9);;
s1 := (1,2)(3,4)(6,7)(8,9);;
s2 := (5,6);;
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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(9)!(2,3)(7,9);
s1 := Sym(9)!(1,2)(3,4)(6,7)(8,9);
s2 := Sym(9)!(5,6);
poly := sub<Sym(9)|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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;
References : None.
to this polytope
Twisty Puzzle