Polytope of Type {6,6}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*1944f
if this polytope has a name.
Group : SmallGroup(1944,2342)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 162, 486, 162
Order of s0s1s2 : 6
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) :
3-fold quotients : {6,6}*648f, {6,6}*648g
9-fold quotients : {6,6}*216a, {6,6}*216b, {6,6}*216d
18-fold quotients : {6,3}*108, {6,6}*108
27-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
54-fold quotients : {3,6}*36, {6,3}*36
81-fold quotients : {2,6}*24, {6,2}*24
162-fold quotients : {2,3}*12, {3,2}*12
243-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*s0*s1> of order 2.
84 facets:
6 of {3}*6
78 of {6}*12
81 vertex figures:
81 of {6}*12
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
81 facets:
81 of {6}*12
90 vertex figures:
72 of {6}*12
18 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
81 facets:
81 of {6}*12
81 vertex figures:
81 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 3.
54 facets:
54 of {6}*12
60 vertex figures:
51 of {6}*12
9 of {2}*4
P/N, where N=<s0*s1*s0*s1> of order 3.
72 facets:
27 of {2}*4
45 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
54 facets:
54 of {6}*12
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 6.
27 facets:
27 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 6.
27 facets:
27 of {6}*12
36 vertex figures:
18 of {6}*12
18 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
30 facets:
6 of {3}*6
24 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 6.
30 facets:
6 of {3}*6
24 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
36 facets:
27 of {2}*4
9 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
18 facets:
18 of {6}*12
24 vertex figures:
15 of {6}*12
9 of {2}*4
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 9.
24 facets:
9 of {2}*4
15 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 9.
30 facets:
18 of {2}*4
12 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
24 facets:
15 of {6}*12
9 of {2}*4
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 9.
24 facets:
15 of {6}*12
9 of {2}*4
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 9.
30 facets:
18 of {2}*4
12 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 9.
24 facets:
9 of {2}*4
15 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 18.
15 facets:
9 of {2}*4
6 of {6}*12
12 vertex figures:
6 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 18.
12 facets:
6 of {3}*6
6 of {6}*12
9 vertex figures:
9 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 27.
8 facets:
5 of {6}*12
3 of {2}*4
6 vertex figures:
6 of {6}*12
P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 27.
14 facets:
12 of {2}*4
2 of {6}*12
6 vertex figures:
6 of {6}*12
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);;
s1 := (10,27)(11,25)(12,26)(13,21)(14,19)(15,20)(16,24)(17,22)(18,23);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27);;
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*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);
s1 := Sym(27)!(10,27)(11,25)(12,26)(13,21)(14,19)(15,20)(16,24)(17,22)(18,23);
s2 := Sym(27)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1 >;
References : None.
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