Polytope of Type {4,10} with group SmallGroup(2000,919)

Overview

Group: SmallGroup(2000,919)
Rank: 3
Schläfli Type: {4,10}
Vertices, edges, …: 100, 500, 250
Order of s₀s₁s₂: 20
Order of s₀s₁s₂s₁: 10
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

5-fold

{4,10}*400

10-fold

{4,10}*200

25-fold

{4,10}*80

50-fold

{2,10}*40

100-fold

{2,5}*20

125-fold

{4,2}*16

250-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*s1)^2*s0*(s1*s2)^2*s1*s0*(s2*s1)^3> of order 2

125 facets

125 of {4}*8

50 vertex figures

50 of {10}*20

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

125 facets

125 of {4}*8

55 vertex figures

45 of {10}*20
10 of {5}*10

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

130 facets

10 of {2}*4
120 of {4}*8

50 vertex figures

50 of {10}*20

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

65 facets

5 of {2}*4
60 of {4}*8

30 vertex figures

20 of {10}*20
10 of {5}*10

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 5

50 facets

50 of {4}*8

40 vertex figures

15 of {10}*20
25 of {2}*4

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

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

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

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

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

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

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

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

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

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*(s1*s2)^3*s1> of order 5

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

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

50 facets

50 of {4}*8

20 vertex figures

20 of {10}*20

P/N, where N=<(s1*s2)^5, s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 10

25 facets

25 of {4}*8

15 vertex figures

10 of {5}*10
5 of {10}*20

P/N, where N=<(s0*s1)^2, s0*(s2*s1)^2*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s2> of order 10

30 facets

10 of {2}*4
20 of {4}*8

10 vertex figures

10 of {10}*20

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2> of order 10

30 facets

10 of {2}*4
20 of {4}*8

10 vertex figures

10 of {10}*20

P/N, where N=<(s0*s1)^2, (s0*s2*s1)^2*s0*s1*s2*s1*s0*s2> of order 10

30 facets

10 of {2}*4
20 of {4}*8

10 vertex figures

10 of {10}*20

P/N, where N=<(s0*s1)^2, (s1*s2)^5> of order 20

15 facets

5 of {2}*4
10 of {4}*8

8 vertex figures

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

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*(s1*s2)^3*s1*s0*s1*s2*s1> of order 25

10 facets

10 of {4}*8

4 vertex figures

4 of {10}*20

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 25

10 facets

10 of {4}*8

8 vertex figures

3 of {10}*20
5 of {2}*4

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, s0*(s1*s2)^3*s1*s0*s1*s2*s1> of order 25

10 facets

10 of {4}*8

8 vertex figures

3 of {10}*20
5 of {2}*4

P/N, where N=<(s1*s0*s1*s2)^2, s0*(s1*s2)^3*s1*s0*s1*s2*s1> of order 25

10 facets

10 of {4}*8

4 vertex figures

4 of {10}*20

P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2> of order 25

10 facets

10 of {4}*8

12 vertex figures

10 of {2}*4
2 of {10}*20

Representations

Permutation Representation (GAP)

s0 := (  6, 26)(  7, 27)(  8, 28)(  9, 29)( 10, 30)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21,101)( 22,102)( 23,103)( 24,104)( 25,105)( 36, 56)( 37, 57)( 38, 58)( 39, 59)( 40, 60)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 85)( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71,111)( 72,112)( 73,113)( 74,114)( 75,115)( 96,116)( 97,117)( 98,118)( 99,119)(100,120);;
s1 := (  2,  5)(  3,  4)(  6, 76)(  7, 80)(  8, 79)(  9, 78)( 10, 77)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 16,101)( 17,105)( 18,104)( 19,103)( 20,102)( 21, 51)( 22, 55)( 23, 54)( 24, 53)( 25, 52)( 31, 86)( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 37, 40)( 38, 39)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46, 61)( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 66,121)( 67,125)( 68,124)( 69,123)( 70,122)( 72, 75)( 73, 74)( 82, 85)( 83, 84)( 91,106)( 92,110)( 93,109)( 94,108)( 95,107)(117,120)(118,119);;
s2 := (  1, 32)(  2, 31)(  3, 35)(  4, 34)(  5, 33)(  6, 27)(  7, 26)(  8, 30)(  9, 29)( 10, 28)( 11, 47)( 12, 46)( 13, 50)( 14, 49)( 15, 48)( 16, 42)( 17, 41)( 18, 45)( 19, 44)( 20, 43)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,102)( 57,101)( 58,105)( 59,104)( 60,103)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)( 66,117)( 67,116)( 68,120)( 69,119)( 70,118)( 71,112)( 72,111)( 73,115)( 74,114)( 75,113)( 76, 82)( 77, 81)( 78, 85)( 79, 84)( 80, 83)( 86, 97)( 87, 96)( 88,100)( 89, 99)( 90, 98)( 91, 92)( 93, 95);;
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*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
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(125)!(  6, 26)(  7, 27)(  8, 28)(  9, 29)( 10, 30)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21,101)( 22,102)( 23,103)( 24,104)( 25,105)( 36, 56)( 37, 57)( 38, 58)( 39, 59)( 40, 60)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 85)( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71,111)( 72,112)( 73,113)( 74,114)( 75,115)( 96,116)( 97,117)( 98,118)( 99,119)(100,120);
s1 := Sym(125)!(  2,  5)(  3,  4)(  6, 76)(  7, 80)(  8, 79)(  9, 78)( 10, 77)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 16,101)( 17,105)( 18,104)( 19,103)( 20,102)( 21, 51)( 22, 55)( 23, 54)( 24, 53)( 25, 52)( 31, 86)( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 37, 40)( 38, 39)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46, 61)( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 66,121)( 67,125)( 68,124)( 69,123)( 70,122)( 72, 75)( 73, 74)( 82, 85)( 83, 84)( 91,106)( 92,110)( 93,109)( 94,108)( 95,107)(117,120)(118,119);
s2 := Sym(125)!(  1, 32)(  2, 31)(  3, 35)(  4, 34)(  5, 33)(  6, 27)(  7, 26)(  8, 30)(  9, 29)( 10, 28)( 11, 47)( 12, 46)( 13, 50)( 14, 49)( 15, 48)( 16, 42)( 17, 41)( 18, 45)( 19, 44)( 20, 43)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,102)( 57,101)( 58,105)( 59,104)( 60,103)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)( 66,117)( 67,116)( 68,120)( 69,119)( 70,118)( 71,112)( 72,111)( 73,115)( 74,114)( 75,113)( 76, 82)( 77, 81)( 78, 85)( 79, 84)( 80, 83)( 86, 97)( 87, 96)( 88,100)( 89, 99)( 90, 98)( 91, 92)( 93, 95);
poly := sub<Sym(125)|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*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
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