Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,4}

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

Overview

Group
SmallGroup(1600,10031)
Rank
4
Schläfli Type
{4,4,4}
Vertices, edges, …
50, 100, 100, 4
Order of s0s1s2s3
20
Order of s0s1s2s3s2s1
2
Also known as
2T4(5,5)(2,0), {{4,4}10,{4,4|2}}. 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=<(s0*s1)^2> of order 2

4 facets

  • 4 of 2-fold non-regular quotient of {4,4}*400

26 vertex figures

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

4 facets

  • 4 of 5-fold non-regular quotient of {4,4}*400

10 vertex figures

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

4 facets

  • 4 of 5-fold non-regular quotient of {4,4}*400

10 vertex figures

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

4 facets

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

6 vertex figures

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

4 facets

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

6 vertex figures

Representations

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

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

to this polytope.