Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*800

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Overview

Group
SmallGroup(800,1058)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
100, 200, 100
Order of s0s1s2
20
Order of s0s1s2s1
10
Also known as
{4,4}(10,0), {4,4|10}. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

4-fold

25-fold

50-fold

100-fold

Covers minimal covers in bold

2-fold

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)^2> of order 2

52 facets

50 vertex figures

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

50 facets

50 vertex figures

P/N, where N=<(s1*s0*s2)^9*s1*s2> of order 2

50 facets

50 vertex figures

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

50 facets

50 vertex figures

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

50 facets

52 vertex figures

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

25 facets

26 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^8*s2*s1*s2> of order 4

26 facets

25 vertex figures

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

25 facets

25 vertex figures

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

20 facets

20 vertex figures

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

20 facets

20 vertex figures

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

20 facets

20 vertex figures

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

10 facets

12 vertex figures

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

10 facets

12 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^4*s1*s0*s2*s1*s2> of order 10

10 facets

10 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s2*s1*s0*s1)^3*s2*s1*s0*s2*s1*s2> of order 10

10 facets

12 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*(s2*s1*s0*s1)^4*s2> of order 10

10 facets

10 vertex figures

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

10 facets

10 vertex figures

P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s0*s2*s1*s0*s1*(s2*s1*s0)^3*s2*s1> of order 10

10 facets

10 vertex figures

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

10 facets

10 vertex figures

P/N, where N=<(s0*s1)^2, s2*s1*s0*s1*(s2*s1*s0)^3*s2*s1*s2> of order 10

12 facets

10 vertex figures

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

12 facets

10 vertex figures

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

12 facets

10 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  2,  7)(  3, 13)(  4, 19)(  5, 25)(  6, 21)(  9, 14)( 10, 20)( 11, 16)( 12, 22)( 18, 23)( 27, 32)( 28, 38)( 29, 44)( 30, 50)( 31, 46)( 34, 39)( 35, 45)( 36, 41)( 37, 47)( 43, 48)( 52, 57)( 53, 63)( 54, 69)( 55, 75)( 56, 71)( 59, 64)( 60, 70)( 61, 66)( 62, 72)( 68, 73)( 77, 82)( 78, 88)( 79, 94)( 80,100)( 81, 96)( 84, 89)( 85, 95)( 86, 91)( 87, 97)( 93, 98);;
s1 := (  2, 19)(  3,  7)(  4, 25)(  5, 13)(  6, 15)(  8, 16)( 10, 22)( 11, 24)( 14, 18)( 17, 21)( 27, 44)( 28, 32)( 29, 50)( 30, 38)( 31, 40)( 33, 41)( 35, 47)( 36, 49)( 39, 43)( 42, 46)( 51, 76)( 52, 94)( 53, 82)( 54,100)( 55, 88)( 56, 90)( 57, 78)( 58, 91)( 59, 84)( 60, 97)( 61, 99)( 62, 87)( 63, 80)( 64, 93)( 65, 81)( 66, 83)( 67, 96)( 68, 89)( 69, 77)( 70, 95)( 71, 92)( 72, 85)( 73, 98)( 74, 86)( 75, 79);;
s2 := (  1, 58)(  2, 52)(  3, 71)(  4, 70)(  5, 64)(  6, 63)(  7, 57)(  8, 51)(  9, 75)( 10, 69)( 11, 68)( 12, 62)( 13, 56)( 14, 55)( 15, 74)( 16, 73)( 17, 67)( 18, 61)( 19, 60)( 20, 54)( 21, 53)( 22, 72)( 23, 66)( 24, 65)( 25, 59)( 26, 83)( 27, 77)( 28, 96)( 29, 95)( 30, 89)( 31, 88)( 32, 82)( 33, 76)( 34,100)( 35, 94)( 36, 93)( 37, 87)( 38, 81)( 39, 80)( 40, 99)( 41, 98)( 42, 92)( 43, 86)( 44, 85)( 45, 79)( 46, 78)( 47, 97)( 48, 91)( 49, 90)( 50, 84);;
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, 
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(100)!(  2,  7)(  3, 13)(  4, 19)(  5, 25)(  6, 21)(  9, 14)( 10, 20)( 11, 16)( 12, 22)( 18, 23)( 27, 32)( 28, 38)( 29, 44)( 30, 50)( 31, 46)( 34, 39)( 35, 45)( 36, 41)( 37, 47)( 43, 48)( 52, 57)( 53, 63)( 54, 69)( 55, 75)( 56, 71)( 59, 64)( 60, 70)( 61, 66)( 62, 72)( 68, 73)( 77, 82)( 78, 88)( 79, 94)( 80,100)( 81, 96)( 84, 89)( 85, 95)( 86, 91)( 87, 97)( 93, 98);
s1 := Sym(100)!(  2, 19)(  3,  7)(  4, 25)(  5, 13)(  6, 15)(  8, 16)( 10, 22)( 11, 24)( 14, 18)( 17, 21)( 27, 44)( 28, 32)( 29, 50)( 30, 38)( 31, 40)( 33, 41)( 35, 47)( 36, 49)( 39, 43)( 42, 46)( 51, 76)( 52, 94)( 53, 82)( 54,100)( 55, 88)( 56, 90)( 57, 78)( 58, 91)( 59, 84)( 60, 97)( 61, 99)( 62, 87)( 63, 80)( 64, 93)( 65, 81)( 66, 83)( 67, 96)( 68, 89)( 69, 77)( 70, 95)( 71, 92)( 72, 85)( 73, 98)( 74, 86)( 75, 79);
s2 := Sym(100)!(  1, 58)(  2, 52)(  3, 71)(  4, 70)(  5, 64)(  6, 63)(  7, 57)(  8, 51)(  9, 75)( 10, 69)( 11, 68)( 12, 62)( 13, 56)( 14, 55)( 15, 74)( 16, 73)( 17, 67)( 18, 61)( 19, 60)( 20, 54)( 21, 53)( 22, 72)( 23, 66)( 24, 65)( 25, 59)( 26, 83)( 27, 77)( 28, 96)( 29, 95)( 30, 89)( 31, 88)( 32, 82)( 33, 76)( 34,100)( 35, 94)( 36, 93)( 37, 87)( 38, 81)( 39, 80)( 40, 99)( 41, 98)( 42, 92)( 43, 86)( 44, 85)( 45, 79)( 46, 78)( 47, 97)( 48, 91)( 49, 90)( 50, 84);
poly := sub<Sym(100)|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, 
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

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