Polytope of Type {6,6} with group SmallGroup(1944,2344)

Overview

Group: SmallGroup(1944,2344)
Rank: 3
Schläfli Type: {6,6}
Vertices, edges, …: 162, 486, 162
Order of s₀s₁s₂: 18
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

6-fold

{6,6}*324b

9-fold

18-fold

{6,6}*108

27-fold

54-fold

81-fold

162-fold

243-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^3*(s2*(s1*s0)^2*s1)^2*s2> of order 2

81 facets

81 of {6}*12

81 vertex figures

81 of {6}*12

P/N, where N=<((s1*s0)^2*s1*s2)^3> 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)^3> 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)^2*(s2*s1)^2*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)^2*s2*s1*s0*(s1*s2)^2*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*s1*s2*s1)^2> 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)^2*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)^2*s2*(s1*s0)^2*(s2*s1)^2*s2> of order 3

54 facets

54 of {6}*12

54 vertex figures

54 of {6}*12

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3

54 facets

54 of {6}*12

72 vertex figures

45 of {6}*12
27 of {2}*4

P/N, where N=<(s0*s1)^3*(s2*s1*s0)^3*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*s0)^2*(s2*s1)^2> of order 3

60 facets

51 of {6}*12
9 of {2}*4

54 vertex figures

54 of {6}*12

P/N, where N=<((s1*s0)^2*s1*s2)^2> 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)^2*s2> of order 3

54 facets

54 of {6}*12

54 vertex figures

54 of {6}*12

P/N, where N=<(s0*s1)^2*(s0*s2*s1)^3*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*s2*s1*s0*(s2*s1)^2*s2, s1*s0*s1*(s2*(s1*s0)^2)^2*s2*s1> of order 6

27 facets

27 of {6}*12

27 vertex figures

27 of {6}*12

P/N, where N=<(s0*s1)^2*s0*(s2*s1*s0*s1)^2*s2*s1> of order 6

27 facets

27 of {6}*12

27 vertex figures

27 of {6}*12

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s0*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*s2*s1*s0*s2*(s1*s0)^2*s2*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=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*(s1*s0)^2*(s2*s1)^2*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=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0> of order 6

27 facets

27 of {6}*12

36 vertex figures

18 of {6}*12
18 of {3}*6

P/N, where N=<(s1*s0)^2*(s2*s1*s0*s1)^2*s2*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=<(s0*s1)^3, s0*s2*(s1*s0)^2*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)^2*s2, (s0*s1)^2*(s2*(s1*s0)^2)^2*s2*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=<(s0*s1)^3, s0*(s1*s2)^2*(s1*s0)^2*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)^3, s1*(s2*s1*s0)^3*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)^3, s0*s1*s2*s1*s0*(s2*s1)^2*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*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s0*s2*s1*s0*(s1*s2)^2*(s1*s0)^2*s2*s1> of order 9

22 facets

16 of {6}*12
6 of {2}*4

18 vertex figures

18 of {6}*12

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*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)^2, ((s1*s0)^2*s1*s2)^2> of order 9

18 facets

18 of {6}*12

18 vertex figures

18 of {6}*12

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*(s2*s1)^2*s2, s0*s1*s2*(s1*s0)^2*s1*s2*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=<s0*s1*s2*s1*s0*s2, (s0*s1)^3*s2*s1*s0*s2*(s1*s0)^2> of order 9

18 facets

18 of {6}*12

30 vertex figures

12 of {6}*12
18 of {2}*4

P/N, where N=<s0*s1*s2*s1*s0*s2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 9

18 facets

18 of {6}*12

24 vertex figures

15 of {6}*12
9 of {2}*4

P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2> of order 9

18 facets

18 of {6}*12

30 vertex figures

18 of {2}*4
12 of {6}*12

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, ((s1*s0)^2*s1*s2)^2> 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*s1*s2*s1)^2, s1*s0*s2*s1*s0*s1*s2*(s1*s0)^2*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*s2*s1*s0*(s2*s1)^2*s2, (s1*s0)^2*s1*s2*(s1*s0)^2*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*s2*s1, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s1*s2*s1> 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*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s1*s2, (s0*s1)^3*(s2*s1*s0)^3*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)^2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 9

18 facets

18 of {6}*12

18 vertex figures

18 of {6}*12

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1, s0*s1*s2*(s1*s0)^2*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=<(s1*s2)^2, ((s1*s0)^2*s1*s2)^2> of order 9

18 facets

18 of {6}*12

24 vertex figures

9 of {2}*4
15 of {6}*12

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s1*s0)^2*(s2*s1)^2*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=<(s0*s1)^2*(s2*s1*s0)^2, s0*s1*s2*s1*s0*(s2*s1)^2*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=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*s2*(s1*s0)^2*s1*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=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0, s0*s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2> 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*(s2*s1)^2*s2, s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1)^2*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)^3, s0*s1*s2*s1*s0*(s2*s1)^2*s2, s0*(s1*s2)^2*(s1*s0)^2*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*s2*s1*s0*s2, s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 18

9 facets

9 of {6}*12

18 vertex figures

3 of {6}*12
9 of {2}*4
6 of {3}*6

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s1*s0*(s2*s1)^2*s0*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)^3, s1*s0*s2*(s1*s0)^2*s1*s2*s1, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0> 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*s2*s1*s0*s2, (s1*s0*s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1> of order 27

6 facets

6 of {6}*12

10 vertex figures

4 of {6}*12
6 of {2}*4

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2, s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 27

6 facets

6 of {6}*12

8 vertex figures

5 of {6}*12
3 of {2}*4

P/N, where N=<(s1*s2)^2, (s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 27

6 facets

6 of {6}*12

12 vertex figures

9 of {2}*4
3 of {6}*12

Representations

Permutation Representation (GAP)

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,33)( 5,32)( 6,31)( 7,35)( 8,34)( 9,36)(10,54)(11,53)(12,52)(13,47)(14,46)(15,48)(16,49)(17,51)(18,50)(19,41)(20,40)(21,42)(22,43)(23,45)(24,44)(25,39)(26,38)(27,37)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)(69,75)(70,76)(71,78)(72,77);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);;
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*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,33)( 5,32)( 6,31)( 7,35)( 8,34)( 9,36)(10,54)(11,53)(12,52)(13,47)(14,46)(15,48)(16,49)(17,51)(18,50)(19,41)(20,40)(21,42)(22,43)(23,45)(24,44)(25,39)(26,38)(27,37)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)(69,75)(70,76)(71,78)(72,77);
s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);
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*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle