Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*648b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(648,299)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
54, 162, 54
Order of s0s1s2
18
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

27-fold

54-fold

81-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)^3> of order 2

30 facets

27 vertex figures

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

18 facets

18 vertex figures

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

18 facets

22 vertex figures

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

30 facets

18 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 6

10 facets

9 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2> of order 6

12 facets

11 vertex figures

P/N, where N=<(s0*s1)^3, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 6

12 facets

9 vertex figures

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

6 facets

6 vertex figures

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

14 facets

6 vertex figures

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

10 facets

6 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s1*s2> of order 18

8 facets

3 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle