Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*576b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(576,8313)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
24, 144, 48
Order of s0s1s2
12
Order of s0s1s2s1
4
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

24-fold

36-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)^2*s2*(s1*s0)^2*s2> of order 2

24 facets

12 vertex figures

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

24 facets

12 vertex figures

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

24 facets

16 vertex figures

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

24 facets

8 vertex figures

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

12 facets

6 vertex figures

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

12 facets

10 vertex figures

Representations

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

References

None.

to this polytope.

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