Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*400

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(400,211)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
50, 100, 50
Order of s0s1s2
10
Order of s0s1s2s1
10
Also known as
{4,4}(5,5), {4,4}10. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

50-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-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*s2*s1)^4*s0*s1*s2> of order 2

25 facets

25 vertex figures

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

26 facets

26 vertex figures

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

13 facets

13 vertex figures

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

10 facets

10 vertex figures

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

10 facets

10 vertex figures

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

6 facets

6 vertex figures

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

6 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle