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}*1152c
if this polytope has a name.
Group : SmallGroup(1152,155790)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 96, 288, 96
Order of s0s1s2 : 12
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}*384c
4-fold quotients : {6,6}*288a
6-fold quotients : {6,6}*192a
8-fold quotients : {6,3}*144
12-fold quotients : {6,6}*96
16-fold quotients : {6,6}*72b
24-fold quotients : {3,6}*48, {6,3}*48
32-fold quotients : {6,3}*36
48-fold quotients : {3,3}*24, {2,6}*24
96-fold quotients : {2,3}*12
144-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*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
48 facets:
48 of {6}*12
48 vertex figures:
48 of {6}*12
P/N, where N=<s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
48 facets:
48 of {6}*12
60 vertex figures:
36 of {6}*12
24 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
52 facets:
8 of {3}*6
44 of {6}*12
48 vertex figures:
48 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 2.
48 facets:
48 of {6}*12
48 vertex figures:
48 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
48 facets:
48 of {6}*12
48 vertex figures:
48 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 2.
48 facets:
48 of {6}*12
48 vertex figures:
48 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
48 facets:
48 of {6}*12
48 vertex figures:
48 of {6}*12
P/N, where N=<s0*s1*s0*s1> of order 3.
36 facets:
6 of {2}*4
30 of {6}*12
32 vertex figures:
32 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
24 facets:
24 of {6}*12
36 vertex figures:
12 of {6}*12
24 of {3}*6
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
26 facets:
4 of {3}*6
22 of {6}*12
30 vertex figures:
18 of {6}*12
12 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
28 facets:
8 of {3}*6
20 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
24 facets:
24 of {6}*12
24 vertex figures:
24 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
12 facets:
12 of {6}*12
18 vertex figures:
6 of {6}*12
12 of {3}*6
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 8.
12 facets:
12 of {6}*12
18 vertex figures:
6 of {6}*12
12 of {3}*6
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
14 facets:
4 of {3}*6
10 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 8.
12 facets:
12 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
14 facets:
4 of {3}*6
10 of {6}*12
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2> of order 12.
12 facets:
6 of {2}*4
6 of {6}*12
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 12.
12 facets:
6 of {2}*4
6 of {6}*12
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 16.
6 facets:
6 of {6}*12
6 vertex figures:
6 of {6}*12
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15)(19,20)(23,24)(25,29)(26,30)(27,32)(28,31)(35,36)(39,40)(41,45)(42,46)(43,48)(44,47);;
s1 := ( 2, 4)( 5,16)( 6,13)( 7,14)( 8,15)( 9,11)(17,33)(18,36)(19,35)(20,34)(21,48)(22,45)(23,46)(24,47)(25,43)(26,42)(27,41)(28,44)(29,38)(30,39)(31,40)(32,37);;
s2 := ( 1,21)( 2,22)( 3,24)( 4,23)( 5,17)( 6,18)( 7,20)( 8,19)( 9,25)(10,26)(11,28)(12,27)(13,29)(14,30)(15,32)(16,31)(33,37)(34,38)(35,40)(36,39)(43,44)(47,48);;
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*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(48)!( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15)(19,20)(23,24)(25,29)(26,30)(27,32)(28,31)(35,36)(39,40)(41,45)(42,46)(43,48)(44,47);
s1 := Sym(48)!( 2, 4)( 5,16)( 6,13)( 7,14)( 8,15)( 9,11)(17,33)(18,36)(19,35)(20,34)(21,48)(22,45)(23,46)(24,47)(25,43)(26,42)(27,41)(28,44)(29,38)(30,39)(31,40)(32,37);
s2 := Sym(48)!( 1,21)( 2,22)( 3,24)( 4,23)( 5,17)( 6,18)( 7,20)( 8,19)( 9,25)(10,26)(11,28)(12,27)(13,29)(14,30)(15,32)(16,31)(33,37)(34,38)(35,40)(36,39)(43,44)(47,48);
poly := sub<Sym(48)|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*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope
Twisty Puzzle