Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,4,4}

Atlas Canonical Name {18,4,4}*1152a

Overview

Group
SmallGroup(1152,99252)
Rank
4
Schläfli Type
{18,4,4}
Vertices, edges, …
18, 72, 16, 8
Order of s0s1s2s3
36
Order of s0s1s2s3s2s1
2
Also known as
{{18,4|2},{4,4}4}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

16-fold

18-fold

24-fold

36-fold

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

6 facets

18 vertex figures

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

Representations

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

References

None.

to this polytope.