Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,4,4}

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

Overview

Group
SmallGroup(1152,99278)
Rank
4
Schläfli Type
{6,4,4}
Vertices, edges, …
18, 72, 48, 8
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=<(s2*s3)^2> of order 2

4 facets

18 vertex figures

  • 18 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<(s0*s1)^2> of order 3

8 facets

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

6 vertex figures

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

8 facets

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

6 vertex figures

P/N, where N=<(s0*s1)^2, (s2*s3)^2> of order 6

4 facets

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

6 vertex figures

  • 6 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<(s2*s3)^2, (s0*s1*s2*s1)^2> of order 6

4 facets

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

6 vertex figures

  • 6 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<(s2*s3)^2, (s1*s0*s1*s2)^2> of order 6

4 facets

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

6 vertex figures

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

Representations

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

References

None.

to this polytope.