Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296o

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,2977)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
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
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

108-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=<(s0*s1)^6> of order 2

30 facets

54 vertex figures

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

18 facets

48 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

15 facets

27 vertex figures

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

12 facets

18 vertex figures

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

12 facets

24 vertex figures

Representations

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