Polytope of Type {6,6} with group SmallGroup(1152,155791)

Overview

Group: SmallGroup(1152,155791)
Rank: 3
Schläfli Type: {6,6}
Vertices, edges, …: 96, 288, 96
Order of s₀s₁s₂: 24
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

{6,6}*384d

4-fold

{6,6}*288b

6-fold

{6,3}*192

8-fold

{3,6}*144

12-fold

{6,6}*96

16-fold

{6,6}*72c

24-fold

32-fold

{3,6}*36

48-fold

96-fold

{3,2}*12

144-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=<((s1*s0)^2*s1*s2)^2> of order 2

48 facets

48 of {6}*12

48 vertex figures

48 of {6}*12

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

48 facets

48 of {6}*12

48 vertex figures

48 of {6}*12

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

48 facets

48 of {6}*12

64 vertex figures

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

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

60 facets

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

48 vertex figures

48 of {6}*12

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

48 facets

48 of {6}*12

48 vertex figures

48 of {6}*12

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

32 facets

32 of {6}*12

36 vertex figures

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

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

36 facets

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

24 vertex figures

24 of {6}*12

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

30 facets

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

32 vertex figures

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

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

24 facets

24 of {6}*12

32 vertex figures

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

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

24 facets

24 of {6}*12

24 vertex figures

24 of {6}*12

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

16 vertex figures

8 of {6}*12
8 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 8

18 facets

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

12 vertex figures

12 of {6}*12

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

18 facets

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

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

12 facets

12 of {6}*12

16 vertex figures

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

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

18 facets

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

16 vertex figures

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

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

12 facets

12 of {6}*12

12 vertex figures

12 of {6}*12

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

6 facets

6 of {6}*12

6 vertex figures

6 of {6}*12

Representations

Permutation Representation (GAP)

s0 := ( 3, 4)( 5, 6)( 9,13)(10,14)(11,16)(12,15)(17,33)(18,34)(19,36)(20,35)(21,38)(22,37)(23,39)(24,40)(25,45)(26,46)(27,48)(28,47)(29,41)(30,42)(31,44)(32,43);;
s1 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,31)( 6,30)( 7,29)( 8,32)( 9,27)(10,26)(11,25)(12,28)(13,23)(14,22)(15,21)(16,24)(34,36)(37,47)(38,46)(39,45)(40,48)(41,43);;
s2 := ( 1, 7)( 2, 8)( 3, 6)( 4, 5)(11,12)(15,16)(17,23)(18,24)(19,22)(20,21)(27,28)(31,32)(33,39)(34,40)(35,38)(36,37)(43,44)(47,48);;
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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*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*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(48)!( 3, 4)( 5, 6)( 9,13)(10,14)(11,16)(12,15)(17,33)(18,34)(19,36)(20,35)(21,38)(22,37)(23,39)(24,40)(25,45)(26,46)(27,48)(28,47)(29,41)(30,42)(31,44)(32,43);
s1 := Sym(48)!( 1,17)( 2,20)( 3,19)( 4,18)( 5,31)( 6,30)( 7,29)( 8,32)( 9,27)(10,26)(11,25)(12,28)(13,23)(14,22)(15,21)(16,24)(34,36)(37,47)(38,46)(39,45)(40,48)(41,43);
s2 := Sym(48)!( 1, 7)( 2, 8)( 3, 6)( 4, 5)(11,12)(15,16)(17,23)(18,24)(19,22)(20,21)(27,28)(31,32)(33,39)(34,40)(35,38)(36,37)(43,44)(47,48);
poly := sub<Sym(48)|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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*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*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