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

Overview

Group: SmallGroup(2000,919)
Rank: 3
Schläfli Type: {10,4}
Vertices, edges, …: 250, 500, 100
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

{10,4}*400

10-fold

{10,4}*200

25-fold

{10,4}*80

50-fold

{10,2}*40

100-fold

{5,2}*20

125-fold

{2,4}*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)^5*s2*(s1*s0)^4*s1*s2> of order 2

50 facets

50 of {10}*20

125 vertex figures

125 of {4}*8

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

55 facets

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

125 vertex figures

125 of {4}*8

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

50 facets

50 of {10}*20

130 vertex figures

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

P/N, where N=<(s0*s1)^5, s0*s2*(s1*s0)^4*s1*s2> of order 4

30 facets

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

65 vertex figures

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

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

20 facets

20 of {10}*20

50 vertex figures

50 of {4}*8

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

20 facets

20 of {10}*20

50 vertex figures

50 of {4}*8

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

20 facets

20 of {10}*20

50 vertex figures

50 of {4}*8

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

20 facets

20 of {10}*20

50 vertex figures

50 of {4}*8

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

40 facets

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

50 vertex figures

50 of {4}*8

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

20 facets

20 of {10}*20

50 vertex figures

50 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 5

20 facets

20 of {10}*20

50 vertex figures

50 of {4}*8

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

20 facets

20 of {10}*20

50 vertex figures

50 of {4}*8

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

15 facets

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

25 vertex figures

25 of {4}*8

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

10 facets

10 of {10}*20

30 vertex figures

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

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

10 facets

10 of {10}*20

30 vertex figures

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

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

10 facets

10 of {10}*20

30 vertex figures

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

P/N, where N=<(s0*s1)^5, s0*s2*(s1*s0)^4*s1*s2, (s0*s1)^2*(s0*s2*s1)^3*s2> of order 20

8 facets

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

15 vertex figures

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

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

4 facets

4 of {10}*20

10 vertex figures

10 of {4}*8

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

8 facets

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

10 vertex figures

10 of {4}*8

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

12 facets

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

10 vertex figures

10 of {4}*8

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

8 facets

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

10 vertex figures

10 of {4}*8

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

4 facets

4 of {10}*20

10 vertex figures

10 of {4}*8

Representations

Permutation Representation (GAP)

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