Polytope of Type {3,15}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,15}*1440
if this polytope has a name.
Group : SmallGroup(1440,5848)
Rank : 3
Schlafli Type : {3,15}
Number of vertices, edges, etc : 48, 360, 240
Order of s0s1s2 : 20
Order of s0s1s2s1 : 15
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) :
4-fold quotients : {3,15}*360
12-fold quotients : {3,5}*120
24-fold quotients : {3,5}*60
60-fold quotients : {3,3}*24
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*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
120 facets:
120 of {3}*6
24 vertex figures:
24 of {15}*30
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2> of order 2.
120 facets:
120 of {3}*6
24 vertex figures:
24 of {15}*30
P/N, where N=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
120 facets:
120 of {3}*6
24 vertex figures:
24 of {15}*30
P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
80 facets:
80 of {3}*6
24 vertex figures:
12 of {15}*30
12 of {5}*10
P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
80 facets:
80 of {3}*6
16 vertex figures:
16 of {15}*30
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
60 facets:
60 of {3}*6
12 vertex figures:
12 of {15}*30
P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
60 facets:
60 of {3}*6
12 vertex figures:
12 of {15}*30
P/N, where N=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1> of order 4.
60 facets:
60 of {3}*6
12 vertex figures:
12 of {15}*30
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
48 facets:
48 of {3}*6
16 vertex figures:
8 of {15}*30
8 of {3}*6
P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 6.
40 facets:
40 of {3}*6
12 vertex figures:
6 of {15}*30
6 of {5}*10
P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 6.
40 facets:
40 of {3}*6
8 vertex figures:
8 of {15}*30
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 10.
24 facets:
24 of {3}*6
8 vertex figures:
4 of {15}*30
4 of {3}*6
P/N, where N=<s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 12.
20 facets:
20 of {3}*6
4 vertex figures:
4 of {15}*30
Permutation Representation (GAP) :
s0 := (2,3)(4,5)(7,9);;
s1 := (1,2)(4,5)(8,9);;
s2 := (2,4)(3,5)(6,8);;
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, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(9)!(2,3)(4,5)(7,9);
s1 := Sym(9)!(1,2)(4,5)(8,9);
s2 := Sym(9)!(2,4)(3,5)(6,8);
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, s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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