Polytope of Type {4,12}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1728e
if this polytope has a name.
Group : SmallGroup(1728,47847)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 72, 432, 216
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
4-fold quotients : {4,12}*432b
9-fold quotients : {4,12}*192c
12-fold quotients : {4,4}*144
18-fold quotients : {4,6}*96
24-fold quotients : {4,4}*72
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
72-fold quotients : {4,3}*24, {2,6}*24
144-fold quotients : {2,3}*12
216-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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 2.
108 facets:
108 of {4}*8
36 vertex figures:
36 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 2.
108 facets:
108 of {4}*8
36 vertex figures:
36 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 2.
108 facets:
108 of {4}*8
36 vertex figures:
36 of {12}*24
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
108 facets:
108 of {4}*8
36 vertex figures:
36 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 2.
108 facets:
108 of {4}*8
40 vertex figures:
32 of {12}*24
8 of {6}*12
P/N, where N=<s0*s1*s0*s1> of order 2.
112 facets:
8 of {2}*4
104 of {4}*8
36 vertex figures:
36 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1> of order 2.
108 facets:
108 of {4}*8
36 vertex figures:
36 of {12}*24
P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
72 facets:
72 of {4}*8
24 vertex figures:
24 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
72 facets:
72 of {4}*8
24 vertex figures:
24 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 3.
72 facets:
72 of {4}*8
24 vertex figures:
24 of {12}*24
P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
72 facets:
72 of {4}*8
36 vertex figures:
18 of {4}*8
18 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
54 facets:
54 of {4}*8
24 vertex figures:
16 of {12}*24
8 of {3}*6
P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
54 facets:
54 of {4}*8
20 vertex figures:
16 of {12}*24
4 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 4.
56 facets:
4 of {2}*4
52 of {4}*8
20 vertex figures:
16 of {12}*24
4 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
54 facets:
54 of {4}*8
18 vertex figures:
18 of {12}*24
P/N, where N=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
54 facets:
54 of {4}*8
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
58 facets:
50 of {4}*8
8 of {2}*4
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
54 facets:
54 of {4}*8
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 4.
56 facets:
52 of {4}*8
4 of {2}*4
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 4.
54 facets:
54 of {4}*8
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 4.
54 facets:
54 of {4}*8
20 vertex figures:
16 of {12}*24
4 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
56 facets:
4 of {2}*4
52 of {4}*8
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
54 facets:
54 of {4}*8
20 vertex figures:
16 of {12}*24
4 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
54 facets:
54 of {4}*8
18 vertex figures:
18 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
36 facets:
36 of {4}*8
18 vertex figures:
9 of {4}*8
9 of {12}*24
P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
36 facets:
36 of {4}*8
18 vertex figures:
9 of {4}*8
9 of {12}*24
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 6.
36 facets:
36 of {4}*8
16 vertex figures:
8 of {12}*24
8 of {6}*12
P/N, where N=<s0*s1*s0*s1, s2*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 6.
40 facets:
8 of {2}*4
32 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
40 facets:
8 of {2}*4
32 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
36 facets:
36 of {4}*8
16 vertex figures:
8 of {6}*12
8 of {12}*24
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
36 facets:
36 of {4}*8
12 vertex figures:
12 of {12}*24
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 8.
28 facets:
2 of {2}*4
26 of {4}*8
10 vertex figures:
8 of {12}*24
2 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
28 facets:
2 of {2}*4
26 of {4}*8
10 vertex figures:
8 of {12}*24
2 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
27 facets:
27 of {4}*8
12 vertex figures:
8 of {12}*24
4 of {3}*6
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
28 facets:
2 of {2}*4
26 of {4}*8
10 vertex figures:
8 of {12}*24
2 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
29 facets:
25 of {4}*8
4 of {2}*4
9 vertex figures:
9 of {12}*24
P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
24 facets:
24 of {4}*8
12 vertex figures:
6 of {4}*8
6 of {12}*24
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 12.
18 facets:
18 of {4}*8
8 vertex figures:
4 of {6}*12
4 of {12}*24
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 12.
20 facets:
4 of {2}*4
16 of {4}*8
6 vertex figures:
6 of {12}*24
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 12.
18 facets:
18 of {4}*8
6 vertex figures:
6 of {12}*24
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
18 facets:
18 of {4}*8
8 vertex figures:
4 of {12}*24
4 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
20 facets:
4 of {2}*4
16 of {4}*8
8 vertex figures:
4 of {12}*24
4 of {6}*12
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
22 facets:
14 of {4}*8
8 of {2}*4
6 vertex figures:
6 of {12}*24
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5,31)( 6,32)( 7,29)( 8,30)( 9,23)(10,24)(11,21)(12,22)(13,27)(14,28)(15,25)(16,26)(17,19)(18,20)(33,35)(34,36);;
s1 := ( 3, 4)( 7, 8)(11,12)(13,33)(14,34)(15,36)(16,35)(17,25)(18,26)(19,28)(20,27)(21,29)(22,30)(23,32)(24,31);;
s2 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,25)( 6,28)( 7,27)( 8,26)(10,12)(13,29)(14,32)(15,31)(16,30)(22,24)(34,36);;
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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*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*s1*s2*s1*s2,
s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(36)!( 1, 3)( 2, 4)( 5,31)( 6,32)( 7,29)( 8,30)( 9,23)(10,24)(11,21)(12,22)(13,27)(14,28)(15,25)(16,26)(17,19)(18,20)(33,35)(34,36);
s1 := Sym(36)!( 3, 4)( 7, 8)(11,12)(13,33)(14,34)(15,36)(16,35)(17,25)(18,26)(19,28)(20,27)(21,29)(22,30)(23,32)(24,31);
s2 := Sym(36)!( 1,17)( 2,20)( 3,19)( 4,18)( 5,25)( 6,28)( 7,27)( 8,26)(10,12)(13,29)(14,32)(15,31)(16,30)(22,24)(34,36);
poly := sub<Sym(36)|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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*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*s1*s2*s1*s2,
s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0 >;
References : None.
to this polytope
Twisty Puzzle