Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*576b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(576,8328)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
48, 144, 48
Order of s0s1s2
24
Order of s0s1s2s1
12
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

4-fold

6-fold

8-fold

12-fold

16-fold

24-fold

48-fold

72-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

36 facets

24 vertex figures

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

24 facets

28 vertex figures

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

20 facets

16 vertex figures

P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 3

16 facets

24 vertex figures

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

12 facets

12 vertex figures

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

14 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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