Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3}

Atlas Canonical Name {6,3}*1728

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Overview

Group
SmallGroup(1728,12317)
Rank
3
Schläfli Type
{6,3}
Vertices, edges, …
288, 432, 144
Order of s0s1s2
24
Order of s0s1s2s1
6
Also known as
{6,3}(12,0), {6,3}24. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

16-fold

36-fold

48-fold

72-fold

144-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=<((s1*s0)^2*s1*s2)^6> of order 2

72 facets

144 vertex figures

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

74 facets

144 vertex figures

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

48 facets

96 vertex figures

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

50 facets

96 vertex figures

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

36 facets

72 vertex figures

P/N, where N=<(s0*s1)^3, s0*(s2*(s1*s0)^2*s1)^5*s2> of order 4

38 facets

72 vertex figures

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

36 facets

72 vertex figures

P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2*s1)^5*s2*s1> of order 4

36 facets

72 vertex figures

P/N, where N=<s1*s0*(s1*s2*(s1*s0)^2)^5*s1*s2*s1> of order 4

36 facets

72 vertex figures

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

26 facets

48 vertex figures

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

24 facets

48 vertex figures

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

24 facets

48 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^3, (s0*s1)^2*((s2*(s1*s0)^2)^2)^2*s2*(s1*s0)^2*s2*s1> of order 8

18 facets

36 vertex figures

P/N, where N=<(s0*s1)^3*(s2*(s1*s0)^2*s1)^2*s2, s0*s1*s0*((s2*(s1*s0)^2)^2)^2*s2*(s1*s0)^2*s2*s1> of order 8

18 facets

36 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^3, (s0*s1)^2*(s2*(s1*s0)^2*s1)^2*s2*s1*s0> of order 8

18 facets

36 vertex figures

P/N, where N=<(s0*s1)^3, s0*(s2*(s1*s0)^2*s1)^5*s2, s0*s1*s0*(s2*(s1*s0)^2*s1)^5*s2*s1> of order 8

19 facets

36 vertex figures

P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2*s1)^2*s2*s1> of order 8

18 facets

36 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^3, s0*s1*s0*(s2*(s1*s0)^2*s1)^2*s2*(s1*s0)^2*s2*s1> of order 12

12 facets

24 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2, s0*s1*s0*((s2*(s1*s0)^2)^2)^2*s2*(s1*s0)^2*s2*s1> of order 12

12 facets

24 vertex figures

P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2*s1*s0*s1> of order 12

12 facets

24 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2> of order 12

14 facets

24 vertex figures

P/N, where N=<(s0*s1)^3, s1*s0*(s2*(s1*s0)^2*s1)^3*s2*s1, s0*(s1*s2*(s1*s0)^2)^2*s1*(s2*(s1*s0)^2)^2*s2*(s1*s0)^2*s2> of order 12

14 facets

24 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2, s0*s1*s0*s2*(s1*s0)^2*s1*((s2*(s1*s0)^2)^2)^2*s2*s1> of order 12

12 facets

24 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 24

6 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle