Polytope of Type {6,20} with group SmallGroup(1920,238598)

Overview

Group: SmallGroup(1920,238598)
Rank: 3
Schläfli Type: {6,20}
Vertices, edges, …: 48, 480, 160
Order of s₀s₁s₂: 30
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

4-fold

{6,20}*480c

5-fold

{6,4}*384a

8-fold

{6,20}*240b

10-fold

{6,4}*192a

16-fold

{6,10}*120

20-fold

{6,4}*96

40-fold

48-fold

{2,10}*40

80-fold

96-fold

{2,5}*20

160-fold

{3,2}*12

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

80 facets

80 of {6}*12

28 vertex figures

20 of {20}*40
8 of {10}*20

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

80 facets

80 of {6}*12

24 vertex figures

24 of {20}*40

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

40 facets

40 of {6}*12

18 vertex figures

12 of {10}*20
6 of {20}*40

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

40 facets

40 of {6}*12

16 vertex figures

8 of {20}*40
8 of {10}*20

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

40 facets

40 of {6}*12

16 vertex figures

8 of {20}*40
8 of {10}*20

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

40 facets

40 of {6}*12

14 vertex figures

10 of {20}*40
4 of {10}*20

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

40 facets

40 of {6}*12

14 vertex figures

10 of {20}*40
4 of {10}*20

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

14 vertex figures

10 of {20}*40
4 of {10}*20

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

40 facets

40 of {6}*12

14 vertex figures

10 of {20}*40
4 of {10}*20

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

14 vertex figures

10 of {20}*40
4 of {10}*20

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

12 vertex figures

12 of {20}*40

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

40 facets

40 of {6}*12

14 vertex figures

10 of {20}*40
4 of {10}*20

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

20 facets

20 of {6}*12

8 vertex figures

4 of {20}*40
4 of {10}*20

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

20 facets

20 of {6}*12

8 vertex figures

4 of {20}*40
4 of {10}*20

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

20 facets

20 of {6}*12

10 vertex figures

2 of {20}*40
8 of {10}*20

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

20 facets

20 of {6}*12

6 vertex figures

6 of {20}*40

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

20 facets

20 of {6}*12

6 vertex figures

6 of {20}*40

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

20 facets

20 of {6}*12

9 vertex figures

6 of {10}*20
3 of {20}*40

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

20 facets

20 of {6}*12

8 vertex figures

4 of {20}*40
4 of {10}*20

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

20 facets

20 of {6}*12

8 vertex figures

4 of {20}*40
4 of {10}*20

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

20 facets

20 of {6}*12

6 vertex figures

6 of {20}*40

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

20 facets

20 of {6}*12

7 vertex figures

5 of {20}*40
2 of {10}*20

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

20 facets

20 of {6}*12

6 vertex figures

6 of {20}*40

Representations

Permutation Representation (GAP)

s0 := ( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15)(19,20)(23,24)(25,29)(26,30)(27,32)(28,31)(35,36)(39,40)(41,45)(42,46)(43,48)(44,47)(51,52)(55,56)(57,61)(58,62)(59,64)(60,63)(67,68)(71,72)(73,77)(74,78)(75,80)(76,79);;
s1 := ( 2, 4)( 5,16)( 6,13)( 7,14)( 8,15)( 9,11)(17,65)(18,68)(19,67)(20,66)(21,80)(22,77)(23,78)(24,79)(25,75)(26,74)(27,73)(28,76)(29,70)(30,71)(31,72)(32,69)(33,49)(34,52)(35,51)(36,50)(37,64)(38,61)(39,62)(40,63)(41,59)(42,58)(43,57)(44,60)(45,54)(46,55)(47,56)(48,53);;
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)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(80)!( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15)(19,20)(23,24)(25,29)(26,30)(27,32)(28,31)(35,36)(39,40)(41,45)(42,46)(43,48)(44,47)(51,52)(55,56)(57,61)(58,62)(59,64)(60,63)(67,68)(71,72)(73,77)(74,78)(75,80)(76,79);
s1 := Sym(80)!( 2, 4)( 5,16)( 6,13)( 7,14)( 8,15)( 9,11)(17,65)(18,68)(19,67)(20,66)(21,80)(22,77)(23,78)(24,79)(25,75)(26,74)(27,73)(28,76)(29,70)(30,71)(31,72)(32,69)(33,49)(34,52)(35,51)(36,50)(37,64)(38,61)(39,62)(40,63)(41,59)(42,58)(43,57)(44,60)(45,54)(46,55)(47,56)(48,53);
s2 := Sym(80)!( 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)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64);
poly := sub<Sym(80)|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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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