Polytope of Type {4,12} with group SmallGroup(1728,47847)

Overview

Group: SmallGroup(1728,47847)
Rank: 3
Schläfli Type: {4,12}
Vertices, edges, …: 72, 432, 216
Order of s₀s₁s₂: 6
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

{4,12}*432b

9-fold

{4,12}*192c

12-fold

{4,4}*144

18-fold

{4,6}*96

24-fold

{4,4}*72

36-fold

72-fold

144-fold

{2,3}*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*s2)^3*s1)^2> of order 2

108 facets

108 of {4}*8

36 vertex figures

36 of {12}*24

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

108 facets

108 of {4}*8

36 vertex figures

36 of {12}*24

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

108 facets

108 of {4}*8

36 vertex figures

36 of {12}*24

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

108 facets

108 of {4}*8

36 vertex figures

36 of {12}*24

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

108 facets

108 of {4}*8

40 vertex figures

32 of {12}*24
8 of {6}*12

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

112 facets

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

36 vertex figures

36 of {12}*24

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

108 facets

108 of {4}*8

36 vertex figures

36 of {12}*24

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

72 facets

72 of {4}*8

24 vertex figures

24 of {12}*24

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

72 facets

72 of {4}*8

24 vertex figures

24 of {12}*24

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

72 facets

72 of {4}*8

24 vertex figures

24 of {12}*24

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

72 facets

72 of {4}*8

36 vertex figures

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

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

54 facets

54 of {4}*8

24 vertex figures

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

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

54 facets

54 of {4}*8

20 vertex figures

16 of {12}*24
4 of {6}*12

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

56 facets

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

20 vertex figures

16 of {12}*24
4 of {6}*12

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

54 facets

54 of {4}*8

18 vertex figures

18 of {12}*24

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

54 facets

54 of {4}*8

18 vertex figures

18 of {12}*24

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

58 facets

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

18 vertex figures

18 of {12}*24

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

54 facets

54 of {4}*8

18 vertex figures

18 of {12}*24

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

56 facets

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

18 vertex figures

18 of {12}*24

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

54 facets

54 of {4}*8

18 vertex figures

18 of {12}*24

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

54 facets

54 of {4}*8

20 vertex figures

16 of {12}*24
4 of {6}*12

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

56 facets

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

18 vertex figures

18 of {12}*24

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

54 facets

54 of {4}*8

20 vertex figures

16 of {12}*24
4 of {6}*12

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

54 facets

54 of {4}*8

18 vertex figures

18 of {12}*24

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

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

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

36 facets

36 of {4}*8

18 vertex figures

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

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

36 facets

36 of {4}*8

18 vertex figures

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

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

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

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

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

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

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

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

36 facets

36 of {4}*8

16 vertex figures

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

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

40 facets

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

12 vertex figures

12 of {12}*24

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

40 facets

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

12 vertex figures

12 of {12}*24

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

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 6

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

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

36 facets

36 of {4}*8

16 vertex figures

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

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

36 facets

36 of {4}*8

12 vertex figures

12 of {12}*24

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

28 facets

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

10 vertex figures

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

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

28 facets

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

10 vertex figures

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

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

27 facets

27 of {4}*8

12 vertex figures

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

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

28 facets

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

10 vertex figures

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

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

29 facets

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

9 vertex figures

9 of {12}*24

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

24 facets

24 of {4}*8

12 vertex figures

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

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

18 facets

18 of {4}*8

8 vertex figures

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

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

20 facets

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

6 vertex figures

6 of {12}*24

P/N, where N=<s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1, s1*s0*(s2*s1)^4*s0*s1*s2> of order 12

18 facets

18 of {4}*8

6 vertex figures

6 of {12}*24

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

18 facets

18 of {4}*8

8 vertex figures

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

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

20 facets

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

8 vertex figures

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

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

22 facets

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

6 vertex figures

6 of {12}*24

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

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