Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,4}

Atlas Canonical Name {18,4}*576a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(576,1380)
Rank
3
Schläfli Type
{18,4}
Vertices, edges, …
72, 144, 16
Order of s0s1s2
18
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable
  • Self-Petrie

Quotients maximal quotients in bold

3-fold

4-fold

8-fold

12-fold

24-fold

Covers minimal covers in bold

2-fold

3-fold

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)^9> of order 2

10 facets

36 vertex figures

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

8 facets

36 vertex figures

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

8 facets

42 vertex figures

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*(s1*s2)^2> of order 4

4 facets

24 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2, (s0*s1)^9> of order 4

6 facets

18 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 4

4 facets

27 vertex figures

Representations

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

Twisty Puzzle