Polytope of Type {6,4} with group SmallGroup(1440,5849)

Overview

Group: SmallGroup(1440,5849)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 180, 360, 120
Order of s₀s₁s₂: 30
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

2-fold

{6,4}*720

3-fold

{6,4}*480

6-fold

12-fold

{6,4}*120

60-fold

{6,2}*24

120-fold

{3,2}*12

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

60 facets

60 of {6}*12

90 vertex figures

90 of {4}*8

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

60 facets

60 of {6}*12

90 vertex figures

90 of {4}*8

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

66 facets

54 of {6}*12
12 of {3}*6

90 vertex figures

90 of {4}*8

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

60 facets

60 of {6}*12

90 vertex figures

90 of {4}*8

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

60 facets

60 of {6}*12

96 vertex figures

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

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

60 facets

60 of {6}*12

90 vertex figures

90 of {4}*8

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

40 facets

40 of {6}*12

60 vertex figures

60 of {4}*8

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

44 facets

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

60 vertex figures

60 of {4}*8

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

30 facets

30 of {6}*12

54 vertex figures

36 of {4}*8
18 of {2}*4

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

30 facets

30 of {6}*12

48 vertex figures

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

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

30 facets

30 of {6}*12

48 vertex figures

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

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

36 facets

24 of {6}*12
12 of {3}*6

48 vertex figures

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

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

33 facets

27 of {6}*12
6 of {3}*6

45 vertex figures

45 of {4}*8

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

30 facets

30 of {6}*12

48 vertex figures

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

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

20 facets

20 of {6}*12

30 vertex figures

30 of {4}*8

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

20 facets

20 of {6}*12

36 vertex figures

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

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

22 facets

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

30 vertex figures

30 of {4}*8

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

26 facets

14 of {6}*12
12 of {3}*6

30 vertex figures

30 of {4}*8

P/N, where N=<(s1*s2*(s1*s0)^2)^2*s1*s2, s0*(s1*s0*s2)^5> of order 6

20 facets

20 of {6}*12

30 vertex figures

30 of {4}*8

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

20 facets

20 of {6}*12

30 vertex figures

30 of {4}*8

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

22 facets

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

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, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 8

18 facets

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

24 vertex figures

21 of {4}*8
3 of {2}*4

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

18 facets

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

27 vertex figures

18 of {4}*8
9 of {2}*4

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

24 vertex figures

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

P/N, where N=<(s0*s1*s2*s1)^3, s0*(s1*s2*(s1*s0)^2)^2*s2> of order 10

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

10 facets

10 of {6}*12

18 vertex figures

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

P/N, where N=<(s0*s1)^3, s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s0*s2> of order 12

14 facets

8 of {3}*6
6 of {6}*12

18 vertex figures

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

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

14 facets

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

18 vertex figures

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

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

6 facets

6 of {6}*12

12 vertex figures

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

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s2, s1*s0*s2*(s1*s0)^2*s1*s2*s1> of order 24

8 facets

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

9 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := ( 1, 2)( 3, 5)( 4, 6)( 9,11);;
s1 := ( 2, 6)( 4, 5)( 7, 8)(10,11);;
s2 := ( 8,10);;
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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(11)!( 1, 2)( 3, 5)( 4, 6)( 9,11);
s1 := Sym(11)!( 2, 6)( 4, 5)( 7, 8)(10,11);
s2 := Sym(11)!( 8,10);
poly := sub<Sym(11)|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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*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