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}*1944h
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 162, 486, 162
Order of s0s1s2 : 18
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}*648b, {6,6}*648g
6-fold quotients : {6,6}*324a
9-fold quotients : {6,6}*216b, {6,6}*216d
18-fold quotients : {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*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*s0*s1> of order 2.
90 facets:
18 of {3}*6
72 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
84 vertex figures:
78 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*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*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=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
54 facets:
54 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> 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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*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=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*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*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1> 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*s2*s1*s0*s2*s1*s2*s1*s0*s2*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*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=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 6.
27 facets:
27 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
27 facets:
27 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*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*s2*s1*s0*s2*s1*s0*s2*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, s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*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.
36 facets:
18 of {3}*6
18 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 6.
36 facets:
18 of {3}*6
18 of {6}*12
27 vertex figures:
27 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*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*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 6.
27 facets:
27 of {6}*12
30 vertex figures:
24 of {6}*12
6 of {3}*6
P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 6.
27 facets:
27 of {6}*12
30 vertex figures:
24 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
27 facets:
27 of {6}*12
30 vertex figures:
24 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 6.
27 facets:
27 of {6}*12
30 vertex figures:
24 of {6}*12
6 of {3}*6
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 9.
18 facets:
18 of {6}*12
22 vertex figures:
16 of {6}*12
6 of {2}*4
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*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*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*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=<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=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 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*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 9.
18 facets:
18 of {6}*12
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
30 facets:
12 of {6}*12
18 of {2}*4
18 vertex figures:
18 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*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=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
30 facets:
12 of {6}*12
18 of {2}*4
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> 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*s2*s1*s0*s1*s2*s1, s1*s0*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*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, 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*s0*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=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*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, s1*s2*s1*s0*s1*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=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 18.
12 facets:
6 of {3}*6
6 of {6}*12
9 vertex figures:
9 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 18.
18 facets:
6 of {3}*6
9 of {2}*4
3 of {6}*12
9 vertex figures:
9 of {6}*12
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 18.
12 facets:
6 of {3}*6
6 of {6}*12
9 vertex figures:
9 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, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 18.
9 facets:
9 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*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> 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*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 18.
9 facets:
9 of {6}*12
12 vertex figures:
6 of {3}*6
6 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 18.
9 facets:
9 of {6}*12
12 vertex figures:
6 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*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, s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1> of order 27.
12 facets:
9 of {2}*4
3 of {6}*12
6 vertex figures:
6 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 27.
10 facets:
4 of {6}*12
6 of {2}*4
6 vertex figures:
6 of {6}*12
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68);;
s1 := ( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,46)(20,48)(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(80,81);;
s2 := ( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)(28,55)(29,56)(30,57)(31,62)(32,63)(33,61)(34,60)(35,58)(36,59)(37,68)(38,69)(39,67)(40,66)(41,64)(42,65)(43,70)(44,71)(45,72)(46,81)(47,79)(48,80)(49,76)(50,77)(51,78)(52,74)(53,75)(54,73);;
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,
s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68);
s1 := Sym(81)!( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,46)(20,48)(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(80,81);
s2 := Sym(81)!( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)(28,55)(29,56)(30,57)(31,62)(32,63)(33,61)(34,60)(35,58)(36,59)(37,68)(38,69)(39,67)(40,66)(41,64)(42,65)(43,70)(44,71)(45,72)(46,81)(47,79)(48,80)(49,76)(50,77)(51,78)(52,74)(53,75)(54,73);
poly := sub<Sym(81)|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,
s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope
Twisty Puzzle