Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,18}

Atlas Canonical Name {4,18}*1296a

▶ Play as a twisty puzzle

Overview

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

Special Properties

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

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=<(s1*s2*s1*s0)^8*s1*s2> of order 2

81 facets

19 vertex figures

P/N, where N=<(s0*s1*s2*s1)^6> of order 3

54 facets

12 vertex figures

P/N, where N=<(s1*s2)^6> of order 3

54 facets

24 vertex figures

P/N, where N=<(s1*s2)^6, s1*s2*s1*s0*(s1*s2)^7*s1*s0*s1*s2> of order 6

27 facets

13 vertex figures

P/N, where N=<(s1*s2)^6, s0*s1*s0*s2*s1*s0*(s1*s2)^7*s1*s0*s1*s2> of order 6

27 facets

13 vertex figures

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

18 facets

4 vertex figures

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

18 facets

4 vertex figures

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

18 facets

20 vertex figures

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

18 facets

8 vertex figures

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

9 facets

11 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle