Polytope of Type {6,4} with group SmallGroup(960,10871)

Overview

Group: SmallGroup(960,10871)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 120, 240, 80
Order of s₀s₁s₂: 20
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Related Polytopes

Constructions

Facet of

{6,4,2} ·1920

Vertex figure of

{2,6,4} ·1920

Quotients maximal quotients in bold

2-fold

{6,4}*480

4-fold

8-fold

{6,4}*120

60-fold

{2,4}*16

120-fold

{2,2}*8

Covers minimal covers in bold

2-fold

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*s2*(s1*s0)^2*s2*s1*s2> of order 2

40 facets

40 of {6}*12

60 vertex figures

60 of {4}*8

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

40 facets

40 of {6}*12

60 vertex figures

60 of {4}*8

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

40 facets

40 of {6}*12

60 vertex figures

60 of {4}*8

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

40 facets

40 of {6}*12

64 vertex figures

56 of {4}*8
8 of {2}*4

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

40 facets

40 of {6}*12

60 vertex figures

60 of {4}*8

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

42 facets

38 of {6}*12
4 of {3}*6

60 vertex figures

60 of {4}*8

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

32 facets

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

40 vertex figures

40 of {4}*8

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 4

20 facets

20 of {6}*12

30 vertex figures

30 of {4}*8

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 4

20 facets

20 of {6}*12

30 vertex figures

30 of {4}*8

P/N, where N=<s0*(s2*s1*s0*s1)^2> of order 4

20 facets

20 of {6}*12

30 vertex figures

30 of {4}*8

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 4

20 facets

20 of {6}*12

34 vertex figures

26 of {4}*8
8 of {2}*4

P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*(s2*s1*s0)^2*s1*s2, (s0*s1)^3*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 4

21 facets

19 of {6}*12
2 of {3}*6

30 vertex figures

30 of {4}*8

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 4

22 facets

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

30 vertex figures

30 of {4}*8

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0, s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 4

22 facets

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

32 vertex figures

28 of {4}*8
4 of {2}*4

P/N, where N=<s0*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2*s1> of order 5

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

16 facets

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

20 vertex figures

20 of {4}*8

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

16 facets

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

20 vertex figures

20 of {4}*8

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

16 facets

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

24 vertex figures

16 of {4}*8
8 of {2}*4

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

18 facets

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

20 vertex figures

20 of {4}*8

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

16 facets

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

20 vertex figures

20 of {4}*8

P/N, where N=<s0*(s2*s1*s0*s1)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 8

10 facets

10 of {6}*12

17 vertex figures

13 of {4}*8
4 of {2}*4

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s1*s2*(s1*s0)^2*s2*s1*s0*s2*s1*s2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 8

11 facets

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

17 vertex figures

13 of {4}*8
4 of {2}*4

P/N, where N=<(s1*s2)^2, s0*s2*s1*s0*s1*(s2*s1*s0)^2*s2> of order 10

8 facets

8 of {6}*12

16 vertex figures

8 of {2}*4
8 of {4}*8

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 10

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2> of order 10

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2> of order 12

12 facets

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

10 vertex figures

10 of {4}*8

P/N, where N=<(s0*s1)^2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s0*s1*s2*(s1*s0)^2*s1*(s2*s1*s0)^2*s1*s2> of order 12

9 facets

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

10 vertex figures

10 of {4}*8

P/N, where N=<s0*(s2*s1*s0*s1)^2, s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 20

4 facets

4 of {6}*12

6 vertex figures

6 of {4}*8

Representations

Permutation Representation (GAP)

s0 := (7,9);;
s1 := (2,3)(5,6)(8,9);;
s2 := (1,2)(3,4)(6,8);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(9)!(7,9);
s1 := Sym(9)!(2,3)(5,6)(8,9);
s2 := Sym(9)!(1,2)(3,4)(6,8);
poly := sub<Sym(9)|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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*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