Polytope of Type {4,15} with group SmallGroup(1920,240395)

Overview

Group: SmallGroup(1920,240395)
Rank: 3
Schläfli Type: {4,15}
Vertices, edges, …: 64, 480, 240
Order of s₀s₁s₂: 15
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

4-fold

{4,15}*480

12-fold

{4,5}*160

16-fold

{4,15}*120

80-fold

{4,3}*24

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

120 facets

120 of {4}*8

32 vertex figures

32 of {15}*30

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

120 facets

120 of {4}*8

32 vertex figures

32 of {15}*30

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

120 facets

120 of {4}*8

32 vertex figures

32 of {15}*30

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

120 facets

120 of {4}*8

32 vertex figures

32 of {15}*30

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

120 facets

120 of {4}*8

32 vertex figures

32 of {15}*30

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

128 facets

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

32 vertex figures

32 of {15}*30

P/N, where N=<(s1*s2)^5> of order 3

80 facets

80 of {4}*8

32 vertex figures

16 of {5}*10
16 of {15}*30

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

64 facets

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

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

68 facets

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

16 vertex figures

16 of {15}*30

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

64 facets

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

16 vertex figures

16 of {15}*30

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

64 facets

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

16 vertex figures

16 of {15}*30

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

64 facets

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

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

68 facets

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

16 vertex figures

16 of {15}*30

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

64 facets

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

16 vertex figures

16 of {15}*30

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

68 facets

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

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

68 facets

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

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

64 facets

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

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

68 facets

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

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

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

60 facets

60 of {4}*8

16 vertex figures

16 of {15}*30

P/N, where N=<(s1*s2)^3> of order 5

48 facets

48 of {4}*8

16 vertex figures

4 of {3}*6
12 of {15}*30

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

40 facets

40 of {4}*8

16 vertex figures

8 of {5}*10
8 of {15}*30

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

32 facets

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

8 vertex figures

8 of {15}*30

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

36 facets

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

8 vertex figures

8 of {15}*30

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

34 facets

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

8 vertex figures

8 of {15}*30

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

36 facets

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

8 vertex figures

8 of {15}*30

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

34 facets

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

8 vertex figures

8 of {15}*30

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

32 facets

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

8 vertex figures

8 of {15}*30

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

30 facets

30 of {4}*8

8 vertex figures

8 of {15}*30

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

34 facets

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

8 vertex figures

8 of {15}*30

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

32 facets

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

8 vertex figures

8 of {15}*30

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

38 facets

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

8 vertex figures

8 of {15}*30

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

30 facets

30 of {4}*8

8 vertex figures

8 of {15}*30

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

30 facets

30 of {4}*8

8 vertex figures

8 of {15}*30

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

36 facets

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

8 vertex figures

8 of {15}*30

Representations

Permutation Representation (GAP)

s0 := ( 1,41)( 2,42)( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56);;
s1 := ( 3, 4)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(19,20)(23,24)(25,30)(26,29)(27,31)(28,32)(33,49)(34,50)(35,52)(36,51)(37,53)(38,54)(39,56)(40,55)(41,62)(42,61)(43,63)(44,64)(45,58)(46,57)(47,59)(48,60);;
s2 := ( 2,10)( 3,11)( 5, 8)( 6,15)( 7,14)(13,16)(17,49)(18,58)(19,59)(20,52)(21,56)(22,63)(23,62)(24,53)(25,57)(26,50)(27,51)(28,60)(29,64)(30,55)(31,54)(32,61)(34,42)(35,43)(37,40)(38,47)(39,46)(45,48);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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, 
s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(64)!( 1,41)( 2,42)( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56);
s1 := Sym(64)!( 3, 4)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(19,20)(23,24)(25,30)(26,29)(27,31)(28,32)(33,49)(34,50)(35,52)(36,51)(37,53)(38,54)(39,56)(40,55)(41,62)(42,61)(43,63)(44,64)(45,58)(46,57)(47,59)(48,60);
s2 := Sym(64)!( 2,10)( 3,11)( 5, 8)( 6,15)( 7,14)(13,16)(17,49)(18,58)(19,59)(20,52)(21,56)(22,63)(23,62)(24,53)(25,57)(26,50)(27,51)(28,60)(29,64)(30,55)(31,54)(32,61)(34,42)(35,43)(37,40)(38,47)(39,46)(45,48);
poly := sub<Sym(64)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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, 
s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2 >;

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