Polytope of Type {6,10}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10}*1440e
if this polytope has a name.
Group : SmallGroup(1440,5849)
Rank : 3
Schlafli Type : {6,10}
Number of vertices, edges, etc : 72, 360, 120
Order of s0s1s2 : 12
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {6,10}*720a
3-fold quotients : {6,10}*480b
6-fold quotients : {6,5}*240a, {6,10}*240a, {6,10}*240b
12-fold quotients : {6,5}*120a
60-fold quotients : {6,2}*24
120-fold quotients : {3,2}*12
180-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=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1> of order 2.
60 facets:
60 of {6}*12
36 vertex figures:
36 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
60 facets:
60 of {6}*12
48 vertex figures:
24 of {10}*20
24 of {5}*10
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1> of order 2.
66 facets:
54 of {6}*12
12 of {3}*6
36 vertex figures:
36 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 2.
60 facets:
60 of {6}*12
36 vertex figures:
36 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
40 facets:
40 of {6}*12
24 vertex figures:
24 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 4.
30 facets:
30 of {6}*12
18 vertex figures:
18 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0> of order 4.
36 facets:
24 of {6}*12
12 of {3}*6
18 vertex figures:
18 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 5.
24 facets:
24 of {6}*12
24 vertex figures:
12 of {10}*20
12 of {2}*4
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 6.
20 facets:
20 of {6}*12
12 vertex figures:
12 of {10}*20
P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 6.
26 facets:
14 of {6}*12
12 of {3}*6
12 vertex figures:
12 of {10}*20
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 10.
12 facets:
12 of {6}*12
12 vertex figures:
6 of {10}*20
6 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 12.
10 facets:
10 of {6}*12
6 vertex figures:
6 of {10}*20
Permutation Representation (GAP) :
s0 := ( 1, 2)( 3, 5)( 4, 6)( 9,11);;
s1 := ( 2, 6)( 4, 5)( 7, 8)(10,11);;
s2 := ( 1, 3)( 2, 5)( 4, 6)( 8,10)( 9,11);;
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*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(11)!( 1, 2)( 3, 5)( 4, 6)( 9,11);
s1 := Sym(11)!( 2, 6)( 4, 5)( 7, 8)(10,11);
s2 := Sym(11)!( 1, 3)( 2, 5)( 4, 6)( 8,10)( 9,11);
poly := sub<Sym(11)|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*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope
Twisty Puzzle