Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,4}

Atlas Canonical Name {18,4}*1296a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,1813)
Rank
3
Schläfli Type
{18,4}
Vertices, edges, …
162, 324, 36
Order of s0s1s2
4
Order of s0s1s2s1
18
Also known as
{18,4}4. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

9-fold

18-fold

81-fold

162-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)^9> of order 2

19 facets

81 vertex figures

P/N, where N=<(s0*s1)^6> of order 3

24 facets

54 vertex figures

P/N, where N=<(s0*s1)^4*s2*(s1*s0)^5*s1*s2*s1*s0*s1> of order 3

12 facets

54 vertex figures

P/N, where N=<(s0*s1)^3> of order 6

13 facets

27 vertex figures

P/N, where N=<(s0*s1)^6, s0*s2*(s1*s0)^8*s1*s2> of order 6

13 facets

27 vertex figures

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

20 facets

18 vertex figures

P/N, where N=<(s0*s1)^6, s0*s1*s2*(s1*s0)^5*s1*s2*s1> of order 9

8 facets

18 vertex figures

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

4 facets

18 vertex figures

P/N, where N=<(s0*s1)^3*s2*s1*s0*s1*s2*s1> of order 9

4 facets

18 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*(s1*s0)^8*s1*s2> of order 18

11 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)(  9, 86)( 10,107)( 11,106)( 12,108)( 13,104)( 14,103)( 15,105)( 16,101)( 17,100)( 18,102)( 19, 98)( 20, 97)( 21, 99)( 22, 95)( 23, 94)( 24, 96)( 25, 92)( 26, 91)( 27, 93)( 28,142)( 29,144)( 30,143)( 31,139)( 32,141)( 33,140)( 34,136)( 35,138)( 36,137)( 37,158)( 38,157)( 39,159)( 40,155)( 41,154)( 42,156)( 43,161)( 44,160)( 45,162)( 46,149)( 47,148)( 48,150)( 49,146)( 50,145)( 51,147)( 52,152)( 53,151)( 54,153)( 55,115)( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61,109)( 62,111)( 63,110)( 64,131)( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,134)( 71,133)( 72,135)( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,125)( 80,124)( 81,126);;
s1 := (  1, 28)(  2, 35)(  3, 33)(  4, 34)(  5, 32)(  6, 30)(  7, 31)(  8, 29)(  9, 36)( 10, 16)( 11, 14)( 15, 18)( 19, 73)( 20, 80)( 21, 78)( 22, 79)( 23, 77)( 24, 75)( 25, 76)( 26, 74)( 27, 81)( 37, 67)( 38, 65)( 39, 72)( 40, 64)( 41, 71)( 42, 69)( 43, 70)( 44, 68)( 45, 66)( 47, 53)( 48, 51)( 49, 52)( 55, 61)( 56, 59)( 60, 63)( 82,109)( 83,116)( 84,114)( 85,115)( 86,113)( 87,111)( 88,112)( 89,110)( 90,117)( 91, 97)( 92, 95)( 96, 99)(100,154)(101,161)(102,159)(103,160)(104,158)(105,156)(106,157)(107,155)(108,162)(118,148)(119,146)(120,153)(121,145)(122,152)(123,150)(124,151)(125,149)(126,147)(128,134)(129,132)(130,133)(136,142)(137,140)(141,144);;
s2 := (  2,  3)(  4,  5)(  7,  9)( 10, 11)( 13, 15)( 17, 18)( 19, 21)( 23, 24)( 25, 26)( 28, 80)( 29, 79)( 30, 81)( 31, 75)( 32, 74)( 33, 73)( 34, 76)( 35, 78)( 36, 77)( 37, 56)( 38, 55)( 39, 57)( 40, 60)( 41, 59)( 42, 58)( 43, 61)( 44, 63)( 45, 62)( 46, 66)( 47, 65)( 48, 64)( 49, 67)( 50, 69)( 51, 68)( 52, 71)( 53, 70)( 54, 72)( 83, 84)( 85, 86)( 88, 90)( 91, 92)( 94, 96)( 98, 99)(100,102)(104,105)(106,107)(109,161)(110,160)(111,162)(112,156)(113,155)(114,154)(115,157)(116,159)(117,158)(118,137)(119,136)(120,138)(121,141)(122,140)(123,139)(124,142)(125,144)(126,143)(127,147)(128,146)(129,145)(130,148)(131,150)(132,149)(133,152)(134,151)(135,153);;
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, 
s2*s0*s1*s2*s0*s1*s2*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*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(162)!(  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)(  9, 86)( 10,107)( 11,106)( 12,108)( 13,104)( 14,103)( 15,105)( 16,101)( 17,100)( 18,102)( 19, 98)( 20, 97)( 21, 99)( 22, 95)( 23, 94)( 24, 96)( 25, 92)( 26, 91)( 27, 93)( 28,142)( 29,144)( 30,143)( 31,139)( 32,141)( 33,140)( 34,136)( 35,138)( 36,137)( 37,158)( 38,157)( 39,159)( 40,155)( 41,154)( 42,156)( 43,161)( 44,160)( 45,162)( 46,149)( 47,148)( 48,150)( 49,146)( 50,145)( 51,147)( 52,152)( 53,151)( 54,153)( 55,115)( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61,109)( 62,111)( 63,110)( 64,131)( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,134)( 71,133)( 72,135)( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,125)( 80,124)( 81,126);
s1 := Sym(162)!(  1, 28)(  2, 35)(  3, 33)(  4, 34)(  5, 32)(  6, 30)(  7, 31)(  8, 29)(  9, 36)( 10, 16)( 11, 14)( 15, 18)( 19, 73)( 20, 80)( 21, 78)( 22, 79)( 23, 77)( 24, 75)( 25, 76)( 26, 74)( 27, 81)( 37, 67)( 38, 65)( 39, 72)( 40, 64)( 41, 71)( 42, 69)( 43, 70)( 44, 68)( 45, 66)( 47, 53)( 48, 51)( 49, 52)( 55, 61)( 56, 59)( 60, 63)( 82,109)( 83,116)( 84,114)( 85,115)( 86,113)( 87,111)( 88,112)( 89,110)( 90,117)( 91, 97)( 92, 95)( 96, 99)(100,154)(101,161)(102,159)(103,160)(104,158)(105,156)(106,157)(107,155)(108,162)(118,148)(119,146)(120,153)(121,145)(122,152)(123,150)(124,151)(125,149)(126,147)(128,134)(129,132)(130,133)(136,142)(137,140)(141,144);
s2 := Sym(162)!(  2,  3)(  4,  5)(  7,  9)( 10, 11)( 13, 15)( 17, 18)( 19, 21)( 23, 24)( 25, 26)( 28, 80)( 29, 79)( 30, 81)( 31, 75)( 32, 74)( 33, 73)( 34, 76)( 35, 78)( 36, 77)( 37, 56)( 38, 55)( 39, 57)( 40, 60)( 41, 59)( 42, 58)( 43, 61)( 44, 63)( 45, 62)( 46, 66)( 47, 65)( 48, 64)( 49, 67)( 50, 69)( 51, 68)( 52, 71)( 53, 70)( 54, 72)( 83, 84)( 85, 86)( 88, 90)( 91, 92)( 94, 96)( 98, 99)(100,102)(104,105)(106,107)(109,161)(110,160)(111,162)(112,156)(113,155)(114,154)(115,157)(116,159)(117,158)(118,137)(119,136)(120,138)(121,141)(122,140)(123,139)(124,142)(125,144)(126,143)(127,147)(128,146)(129,145)(130,148)(131,150)(132,149)(133,152)(134,151)(135,153);
poly := sub<Sym(162)|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, 
s2*s0*s1*s2*s0*s1*s2*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*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle