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}*1944c
if this polytope has a name.
Group : SmallGroup(1944,2324)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 162, 486, 81
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie
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) :
9-fold quotients : {12,6}*216c
27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1> of order 3.
27 facets:
27 of {12}*24
72 vertex figures:
45 of {6}*12
27 of {2}*4
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
27 facets:
27 of {12}*24
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 3.
27 facets:
27 of {12}*24
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
27 facets:
27 of {12}*24
54 vertex figures:
54 of {6}*12
P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 3.
27 facets:
27 of {12}*24
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 3.
27 facets:
27 of {12}*24
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
33 facets:
9 of {4}*8
24 of {12}*24
54 vertex figures:
54 of {6}*12
P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 9.
9 facets:
9 of {12}*24
36 vertex figures:
27 of {2}*4
9 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 9.
9 facets:
9 of {12}*24
30 vertex figures:
12 of {6}*12
18 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 9.
9 facets:
9 of {12}*24
24 vertex figures:
15 of {6}*12
9 of {2}*4
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 9.
9 facets:
9 of {12}*24
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 9.
9 facets:
9 of {12}*24
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 9.
9 facets:
9 of {12}*24
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
9 facets:
9 of {12}*24
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 9.
15 facets:
9 of {4}*8
6 of {12}*24
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
15 facets:
9 of {4}*8
6 of {12}*24
18 vertex figures:
18 of {6}*12
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 5)( 7, 9)(10,19)(11,21)(12,20)(13,23)(14,22)(15,24)(16,27)(17,26)(18,25);;
s1 := ( 1,10)( 2,17)( 3,15)( 4,16)( 5,14)( 6,12)( 7,13)( 8,11)( 9,18)(20,26)(21,24)(22,25);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,11)(13,17)(14,16)(15,18)(19,21)(22,27)(23,26)(24,25);;
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*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(27)!( 2, 3)( 4, 5)( 7, 9)(10,19)(11,21)(12,20)(13,23)(14,22)(15,24)(16,27)(17,26)(18,25);
s1 := Sym(27)!( 1,10)( 2,17)( 3,15)( 4,16)( 5,14)( 6,12)( 7,13)( 8,11)( 9,18)(20,26)(21,24)(22,25);
s2 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,11)(13,17)(14,16)(15,18)(19,21)(22,27)(23,26)(24,25);
poly := sub<Sym(27)|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*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;
References : None.
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