Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*576a

▶ Play as a twisty puzzle

Overview

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

4-fold

6-fold

8-fold

12-fold

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

12 facets

24 vertex figures

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

12 facets

24 vertex figures

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

8 facets

24 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s0*s1)^3*(s0*s2*s1)^3> of order 4

6 facets

12 vertex figures

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

6 facets

12 vertex figures

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

6 facets

12 vertex figures

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

6 facets

12 vertex figures

Representations

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