Polytope of Type {6,4} with group SmallGroup(1728,47847)

Overview

Group: SmallGroup(1728,47847)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 216, 432, 144
Order of s₀s₁s₂: 12
Order of s₀s₁s₂s₁: 12
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

4-fold

{6,4}*432b

9-fold

{6,4}*192b

12-fold

{6,4}*144

18-fold

{6,4}*96

24-fold

{6,4}*72

36-fold

72-fold

108-fold

{2,4}*16

144-fold

{3,2}*12

216-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)^2*s2*(s1*s0)^2*(s2*s1*s0*s1)^2*s2> of order 2

72 facets

72 of {6}*12

108 vertex figures

108 of {4}*8

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

72 facets

72 of {6}*12

108 vertex figures

108 of {4}*8

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

84 facets

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

108 vertex figures

108 of {4}*8

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

72 facets

72 of {6}*12

108 vertex figures

108 of {4}*8

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

72 facets

72 of {6}*12

108 vertex figures

108 of {4}*8

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

72 facets

72 of {6}*12

112 vertex figures

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

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

72 facets

72 of {6}*12

108 vertex figures

108 of {4}*8

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

54 facets

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

72 vertex figures

72 of {4}*8

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

48 facets

48 of {6}*12

72 vertex figures

72 of {4}*8

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

48 facets

48 of {6}*12

72 vertex figures

72 of {4}*8

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

48 facets

48 of {6}*12

72 vertex figures

72 of {4}*8

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

48 facets

48 of {6}*12

72 vertex figures

72 of {4}*8

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

36 facets

36 of {6}*12

56 vertex figures

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

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

48 facets

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

54 vertex figures

54 of {4}*8

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

42 facets

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

56 vertex figures

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

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

56 vertex figures

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

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

58 vertex figures

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

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

42 facets

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

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

36 facets

36 of {6}*12

54 vertex figures

54 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

36 facets

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

36 vertex figures

36 of {4}*8

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 6

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

40 vertex figures

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

P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1*s2, (s0*s1*s2*s1)^4> of order 6

24 facets

24 of {6}*12

40 vertex figures

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

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

28 facets

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

36 vertex figures

36 of {4}*8

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

28 facets

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

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

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

24 facets

24 of {6}*12

36 vertex figures

36 of {4}*8

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1> of order 8

24 facets

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

27 vertex figures

27 of {4}*8

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

18 facets

18 of {6}*12

28 vertex figures

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

P/N, where N=<(s0*s1)^3, s1*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 8

21 facets

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

27 vertex figures

27 of {4}*8

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

18 facets

18 of {6}*12

28 vertex figures

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

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s1*s2, (s1*s0)^2*s2*s1*s0*s1*s2*s1*s0*(s1*s2)^2> of order 8

18 facets

18 of {6}*12

27 vertex figures

27 of {4}*8

P/N, where N=<(s0*s1)^3, (s1*s2*s1*s0)^2*(s1*s2)^2, (s0*s1*s2*s1)^3*s0*s2> of order 8

21 facets

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

29 vertex figures

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

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1*s2> of order 8

24 facets

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

28 vertex figures

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

P/N, where N=<(s0*s1)^3, (s0*s1*s2*s1)^2*s0*(s1*s2)^2> of order 8

21 facets

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

27 vertex figures

27 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^3, ((s1*s0)^2*s1*s2)^2> of order 8

18 facets

18 of {6}*12

28 vertex figures

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

P/N, where N=<(s1*s2*s1*s0)^2*(s1*s2)^2, (s0*s1)^3*s2*(s1*s0)^2*s1*s2> of order 8

18 facets

18 of {6}*12

29 vertex figures

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

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

18 facets

18 of {6}*12

28 vertex figures

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

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, (s0*s1*s2*s1)^4> of order 9

20 facets

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

24 vertex figures

24 of {4}*8

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

20 facets

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

24 vertex figures

24 of {4}*8

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

18 facets

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

24 vertex figures

24 of {4}*8

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

18 facets

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

18 vertex figures

18 of {4}*8

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

20 facets

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

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

20 vertex figures

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

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

12 facets

12 of {6}*12

22 vertex figures

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

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

14 facets

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

20 vertex figures

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

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

18 facets

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

20 vertex figures

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

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

20 vertex figures

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

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

12 facets

12 of {6}*12

22 vertex figures

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

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, (s0*s1*s2*s1)^2*s0*(s1*s2)^2> of order 16

12 facets

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

14 vertex figures

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

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

14 facets

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

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2, (s0*s1)^3*s2*(s1*s0)^2*s1*s2> of order 24

6 facets

6 of {6}*12

11 vertex figures

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

P/N, where N=<(s0*s1)^3, s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s0*s1*s2*(s1*s0)^2*s2*s1*s0*s2*s1*s2> of order 24

9 facets

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

9 vertex figures

9 of {4}*8

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, (s0*s2*s1)^3*s2> of order 24

10 facets

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

10 vertex figures

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

P/N, where N=<(s0*s1)^3, s1*s2*(s1*s0)^2*s2*s1*s0*s2*s1*s2, s0*s1*s2*(s1*s0)^2*(s2*s1*s0)^2*s2> of order 24

9 facets

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

11 vertex figures

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

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

6 facets

6 of {6}*12

10 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);;
s1 := ( 1,13)( 2,16)( 3,15)( 4,14)( 6, 8)( 9,33)(10,36)(11,35)(12,34)(17,29)(18,32)(19,31)(20,30)(22,24)(26,28);;
s2 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,34)(14,33)(15,36)(16,35)(17,26)(18,25)(19,28)(20,27)(21,30)(22,29)(23,32)(24,31);;
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);
s1 := Sym(36)!( 1,13)( 2,16)( 3,15)( 4,14)( 6, 8)( 9,33)(10,36)(11,35)(12,34)(17,29)(18,32)(19,31)(20,30)(22,24)(26,28);
s2 := Sym(36)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,34)(14,33)(15,36)(16,35)(17,26)(18,25)(19,28)(20,27)(21,30)(22,29)(23,32)(24,31);
poly := sub<Sym(36)|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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*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