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