Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,4}

Atlas Canonical Name {4,4,4}*1600a

Overview

Group
SmallGroup(1600,10031)
Rank
4
Schläfli Type
{4,4,4}
Vertices, edges, …
4, 100, 100, 50
Order of s0s1s2s3
20
Order of s0s1s2s3s2s1
2
Also known as
2T4(2,0)(5,5), {{4,4|2},{4,4}10}. if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

50-fold

100-fold

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=<(s1*s2)^2> of order 2

26 facets

4 vertex figures

  • 4 of 2-fold non-regular quotient of {4,4}*400
P/N, where N=<s1*s2*s1*s3*s2*s1*s2*s3*s2*s1*(s2*s3)^2> of order 5

10 facets

4 vertex figures

  • 4 of 5-fold non-regular quotient of {4,4}*400
P/N, where N=<(s1*s2*s3*s2)^2> of order 5

10 facets

4 vertex figures

  • 4 of 5-fold non-regular quotient of {4,4}*400
P/N, where N=<(s1*s2)^2, (s1*s2*s3*s2)^2> of order 10

6 facets

4 vertex figures

  • 4 of 10-fold non-regular quotient of {4,4}*400
P/N, where N=<(s1*s2)^2, s1*s3*s2*s1*s2*(s3*s2*s1)^2*s3> of order 10

6 facets

4 vertex figures

  • 4 of 10-fold non-regular quotient of {4,4}*400

Representations

Permutation Representation (GAP) 
s0 := ( 51, 76)( 52, 77)( 53, 78)( 54, 79)( 55, 80)( 56, 81)( 57, 82)( 58, 83)( 59, 84)( 60, 85)( 61, 86)( 62, 87)( 63, 88)( 64, 89)( 65, 90)( 66, 91)( 67, 92)( 68, 93)( 69, 94)( 70, 95)( 71, 96)( 72, 97)( 73, 98)( 74, 99)( 75,100);;
s1 := (  1, 51)(  2, 62)(  3, 73)(  4, 59)(  5, 70)(  6, 71)(  7, 57)(  8, 68)(  9, 54)( 10, 65)( 11, 66)( 12, 52)( 13, 63)( 14, 74)( 15, 60)( 16, 61)( 17, 72)( 18, 58)( 19, 69)( 20, 55)( 21, 56)( 22, 67)( 23, 53)( 24, 64)( 25, 75)( 26, 76)( 27, 87)( 28, 98)( 29, 84)( 30, 95)( 31, 96)( 32, 82)( 33, 93)( 34, 79)( 35, 90)( 36, 91)( 37, 77)( 38, 88)( 39, 99)( 40, 85)( 41, 86)( 42, 97)( 43, 83)( 44, 94)( 45, 80)( 46, 81)( 47, 92)( 48, 78)( 49, 89)( 50,100);;
s2 := (  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)( 15, 17)( 19, 21)( 27, 34)( 28, 37)( 29, 45)( 30, 48)( 31, 38)( 32, 41)( 33, 49)( 36, 50)( 40, 42)( 44, 46)( 52, 59)( 53, 62)( 54, 70)( 55, 73)( 56, 63)( 57, 66)( 58, 74)( 61, 75)( 65, 67)( 69, 71)( 77, 84)( 78, 87)( 79, 95)( 80, 98)( 81, 88)( 82, 91)( 83, 99)( 86,100)( 90, 92)( 94, 96);;
s3 := (  1,  7)(  2, 21)(  3, 15)(  5, 18)(  6, 12)(  8, 20)( 10, 23)( 11, 17)( 13, 25)( 16, 22)( 26, 32)( 27, 46)( 28, 40)( 30, 43)( 31, 37)( 33, 45)( 35, 48)( 36, 42)( 38, 50)( 41, 47)( 51, 57)( 52, 71)( 53, 65)( 55, 68)( 56, 62)( 58, 70)( 60, 73)( 61, 67)( 63, 75)( 66, 72)( 76, 82)( 77, 96)( 78, 90)( 80, 93)( 81, 87)( 83, 95)( 85, 98)( 86, 92)( 88,100)( 91, 97);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(100)!( 51, 76)( 52, 77)( 53, 78)( 54, 79)( 55, 80)( 56, 81)( 57, 82)( 58, 83)( 59, 84)( 60, 85)( 61, 86)( 62, 87)( 63, 88)( 64, 89)( 65, 90)( 66, 91)( 67, 92)( 68, 93)( 69, 94)( 70, 95)( 71, 96)( 72, 97)( 73, 98)( 74, 99)( 75,100);
s1 := Sym(100)!(  1, 51)(  2, 62)(  3, 73)(  4, 59)(  5, 70)(  6, 71)(  7, 57)(  8, 68)(  9, 54)( 10, 65)( 11, 66)( 12, 52)( 13, 63)( 14, 74)( 15, 60)( 16, 61)( 17, 72)( 18, 58)( 19, 69)( 20, 55)( 21, 56)( 22, 67)( 23, 53)( 24, 64)( 25, 75)( 26, 76)( 27, 87)( 28, 98)( 29, 84)( 30, 95)( 31, 96)( 32, 82)( 33, 93)( 34, 79)( 35, 90)( 36, 91)( 37, 77)( 38, 88)( 39, 99)( 40, 85)( 41, 86)( 42, 97)( 43, 83)( 44, 94)( 45, 80)( 46, 81)( 47, 92)( 48, 78)( 49, 89)( 50,100);
s2 := Sym(100)!(  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)( 15, 17)( 19, 21)( 27, 34)( 28, 37)( 29, 45)( 30, 48)( 31, 38)( 32, 41)( 33, 49)( 36, 50)( 40, 42)( 44, 46)( 52, 59)( 53, 62)( 54, 70)( 55, 73)( 56, 63)( 57, 66)( 58, 74)( 61, 75)( 65, 67)( 69, 71)( 77, 84)( 78, 87)( 79, 95)( 80, 98)( 81, 88)( 82, 91)( 83, 99)( 86,100)( 90, 92)( 94, 96);
s3 := Sym(100)!(  1,  7)(  2, 21)(  3, 15)(  5, 18)(  6, 12)(  8, 20)( 10, 23)( 11, 17)( 13, 25)( 16, 22)( 26, 32)( 27, 46)( 28, 40)( 30, 43)( 31, 37)( 33, 45)( 35, 48)( 36, 42)( 38, 50)( 41, 47)( 51, 57)( 52, 71)( 53, 65)( 55, 68)( 56, 62)( 58, 70)( 60, 73)( 61, 67)( 63, 75)( 66, 72)( 76, 82)( 77, 96)( 78, 90)( 80, 93)( 81, 87)( 83, 95)( 85, 98)( 86, 92)( 88,100)( 91, 97);
poly := sub<Sym(100)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

References

  1. Theorem 10C2, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.