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

Overview

Group: SmallGroup(1920,240395)
Rank: 3
Schläfli Type: {15,4}
Vertices, edges, …: 240, 480, 64
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
Self-Petrie

Quotients maximal quotients in bold

4-fold

{15,4}*480

12-fold

{5,4}*160

16-fold

{15,4}*120

80-fold

{3,4}*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=<s0*s1*s2*s1*s0*(s1*s2)^2> of order 2

32 facets

32 of {15}*30

120 vertex figures

120 of {4}*8

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

32 facets

32 of {15}*30

128 vertex figures

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

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

32 facets

32 of {15}*30

120 vertex figures

120 of {4}*8

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

32 facets

32 of {15}*30

120 vertex figures

120 of {4}*8

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

32 facets

32 of {15}*30

120 vertex figures

120 of {4}*8

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

32 facets

32 of {15}*30

120 vertex figures

120 of {4}*8

P/N, where N=<(s0*s1)^5> of order 3

32 facets

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

80 vertex figures

80 of {4}*8

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

16 facets

16 of {15}*30

64 vertex figures

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

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

16 facets

16 of {15}*30

68 vertex figures

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

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

16 facets

16 of {15}*30

64 vertex figures

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

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

16 facets

16 of {15}*30

64 vertex figures

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

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

16 facets

16 of {15}*30

68 vertex figures

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

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

16 facets

16 of {15}*30

68 vertex figures

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

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

16 facets

16 of {15}*30

68 vertex figures

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

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

64 vertex figures

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

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

16 facets

16 of {15}*30

68 vertex figures

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

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 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*s2*s1> of order 4

16 facets

16 of {15}*30

64 vertex figures

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

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

64 vertex figures

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

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

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

16 facets

16 of {15}*30

60 vertex figures

60 of {4}*8

P/N, where N=<(s0*s1)^3> of order 5

16 facets

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

48 vertex figures

48 of {4}*8

P/N, where N=<(s0*s1)^5, s0*s2*(s1*s0)^3*s2*s1*s0*s1> of order 6

16 facets

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

40 vertex figures

40 of {4}*8

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

8 facets

8 of {15}*30

38 vertex figures

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

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

8 facets

8 of {15}*30

34 vertex figures

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

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

8 facets

8 of {15}*30

36 vertex figures

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

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

8 facets

8 of {15}*30

32 vertex figures

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

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

8 facets

8 of {15}*30

32 vertex figures

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

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

8 facets

8 of {15}*30

34 vertex figures

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

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

8 facets

8 of {15}*30

30 vertex figures

30 of {4}*8

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

8 facets

8 of {15}*30

34 vertex figures

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

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

8 facets

8 of {15}*30

36 vertex figures

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

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

8 facets

8 of {15}*30

32 vertex figures

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

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

8 facets

8 of {15}*30

30 vertex figures

30 of {4}*8

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

8 facets

8 of {15}*30

30 vertex figures

30 of {4}*8

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

8 facets

8 of {15}*30

36 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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