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