Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,6}

Atlas Canonical Name {4,4,6}*1152a

Overview

Group
SmallGroup(1152,99278)
Rank
4
Schläfli Type
{4,4,6}
Vertices, edges, …
8, 48, 72, 18
Order of s0s1s2s3
4
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

9-fold

18-fold

36-fold

72-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

18 facets

  • 18 of 2-fold non-regular quotient of {4,4}*64

4 vertex figures

P/N, where N=<(s2*s3)^2> of order 3

6 facets

8 vertex figures

  • 8 of 3-fold non-regular quotient of {4,6}*144
P/N, where N=<(s2*s1*s2*s3)^2> of order 3

6 facets

8 vertex figures

  • 8 of 3-fold non-regular quotient of {4,6}*144
P/N, where N=<(s0*s1)^2, (s2*s3)^2> of order 6

6 facets

  • 6 of 2-fold non-regular quotient of {4,4}*64

4 vertex figures

  • 4 of 3-fold non-regular quotient of {4,6}*144
P/N, where N=<(s0*s1)^2, (s2*s1*s2*s3)^2> of order 6

6 facets

  • 6 of 2-fold non-regular quotient of {4,4}*64

4 vertex figures

  • 4 of 3-fold non-regular quotient of {4,6}*144
P/N, where N=<(s0*s1)^2, (s1*s2*s3*s2)^2> of order 6

6 facets

  • 6 of 2-fold non-regular quotient of {4,4}*64

4 vertex figures

  • 4 of 3-fold non-regular quotient of {4,6}*144

Representations

Permutation Representation (GAP) 
s0 := ( 19, 28)( 20, 29)( 21, 30)( 22, 31)( 23, 32)( 24, 33)( 25, 34)( 26, 35)( 27, 36)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141)(133,142)(134,143)(135,144);;
s1 := (  2,  6)(  3,  8)(  5,  9)( 11, 15)( 12, 17)( 14, 18)( 20, 24)( 21, 26)( 23, 27)( 29, 33)( 30, 35)( 32, 36)( 38, 42)( 39, 44)( 41, 45)( 47, 51)( 48, 53)( 50, 54)( 56, 60)( 57, 62)( 59, 63)( 65, 69)( 66, 71)( 68, 72)( 73, 91)( 74, 96)( 75, 98)( 76, 94)( 77, 99)( 78, 92)( 79, 97)( 80, 93)( 81, 95)( 82,100)( 83,105)( 84,107)( 85,103)( 86,108)( 87,101)( 88,106)( 89,102)( 90,104)(109,127)(110,132)(111,134)(112,130)(113,135)(114,128)(115,133)(116,129)(117,131)(118,136)(119,141)(120,143)(121,139)(122,144)(123,137)(124,142)(125,138)(126,140);;
s2 := (  1, 73)(  2, 75)(  3, 74)(  4, 77)(  5, 76)(  6, 78)(  7, 81)(  8, 80)(  9, 79)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)( 17, 89)( 18, 88)( 19, 91)( 20, 93)( 21, 92)( 22, 95)( 23, 94)( 24, 96)( 25, 99)( 26, 98)( 27, 97)( 28,100)( 29,102)( 30,101)( 31,104)( 32,103)( 33,105)( 34,108)( 35,107)( 36,106)( 37,109)( 38,111)( 39,110)( 40,113)( 41,112)( 42,114)( 43,117)( 44,116)( 45,115)( 46,118)( 47,120)( 48,119)( 49,122)( 50,121)( 51,123)( 52,126)( 53,125)( 54,124)( 55,127)( 56,129)( 57,128)( 58,131)( 59,130)( 60,132)( 61,135)( 62,134)( 63,133)( 64,136)( 65,138)( 66,137)( 67,140)( 68,139)( 69,141)( 70,144)( 71,143)( 72,142);;
s3 := (  1, 40)(  2, 42)(  3, 41)(  4, 37)(  5, 39)(  6, 38)(  7, 43)(  8, 45)(  9, 44)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 52)( 17, 54)( 18, 53)( 19, 58)( 20, 60)( 21, 59)( 22, 55)( 23, 57)( 24, 56)( 25, 61)( 26, 63)( 27, 62)( 28, 67)( 29, 69)( 30, 68)( 31, 64)( 32, 66)( 33, 65)( 34, 70)( 35, 72)( 36, 71)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,115)( 80,117)( 81,116)( 82,121)( 83,123)( 84,122)( 85,118)( 86,120)( 87,119)( 88,124)( 89,126)( 90,125)( 91,130)( 92,132)( 93,131)( 94,127)( 95,129)( 96,128)( 97,133)( 98,135)( 99,134)(100,139)(101,141)(102,140)(103,136)(104,138)(105,137)(106,142)(107,144)(108,143);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(144)!( 19, 28)( 20, 29)( 21, 30)( 22, 31)( 23, 32)( 24, 33)( 25, 34)( 26, 35)( 27, 36)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141)(133,142)(134,143)(135,144);
s1 := Sym(144)!(  2,  6)(  3,  8)(  5,  9)( 11, 15)( 12, 17)( 14, 18)( 20, 24)( 21, 26)( 23, 27)( 29, 33)( 30, 35)( 32, 36)( 38, 42)( 39, 44)( 41, 45)( 47, 51)( 48, 53)( 50, 54)( 56, 60)( 57, 62)( 59, 63)( 65, 69)( 66, 71)( 68, 72)( 73, 91)( 74, 96)( 75, 98)( 76, 94)( 77, 99)( 78, 92)( 79, 97)( 80, 93)( 81, 95)( 82,100)( 83,105)( 84,107)( 85,103)( 86,108)( 87,101)( 88,106)( 89,102)( 90,104)(109,127)(110,132)(111,134)(112,130)(113,135)(114,128)(115,133)(116,129)(117,131)(118,136)(119,141)(120,143)(121,139)(122,144)(123,137)(124,142)(125,138)(126,140);
s2 := Sym(144)!(  1, 73)(  2, 75)(  3, 74)(  4, 77)(  5, 76)(  6, 78)(  7, 81)(  8, 80)(  9, 79)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)( 17, 89)( 18, 88)( 19, 91)( 20, 93)( 21, 92)( 22, 95)( 23, 94)( 24, 96)( 25, 99)( 26, 98)( 27, 97)( 28,100)( 29,102)( 30,101)( 31,104)( 32,103)( 33,105)( 34,108)( 35,107)( 36,106)( 37,109)( 38,111)( 39,110)( 40,113)( 41,112)( 42,114)( 43,117)( 44,116)( 45,115)( 46,118)( 47,120)( 48,119)( 49,122)( 50,121)( 51,123)( 52,126)( 53,125)( 54,124)( 55,127)( 56,129)( 57,128)( 58,131)( 59,130)( 60,132)( 61,135)( 62,134)( 63,133)( 64,136)( 65,138)( 66,137)( 67,140)( 68,139)( 69,141)( 70,144)( 71,143)( 72,142);
s3 := Sym(144)!(  1, 40)(  2, 42)(  3, 41)(  4, 37)(  5, 39)(  6, 38)(  7, 43)(  8, 45)(  9, 44)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 52)( 17, 54)( 18, 53)( 19, 58)( 20, 60)( 21, 59)( 22, 55)( 23, 57)( 24, 56)( 25, 61)( 26, 63)( 27, 62)( 28, 67)( 29, 69)( 30, 68)( 31, 64)( 32, 66)( 33, 65)( 34, 70)( 35, 72)( 36, 71)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,115)( 80,117)( 81,116)( 82,121)( 83,123)( 84,122)( 85,118)( 86,120)( 87,119)( 88,124)( 89,126)( 90,125)( 91,130)( 92,132)( 93,131)( 94,127)( 95,129)( 96,128)( 97,133)( 98,135)( 99,134)(100,139)(101,141)(102,140)(103,136)(104,138)(105,137)(106,142)(107,144)(108,143);
poly := sub<Sym(144)|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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.