Polytope of Type {3,12} with group SmallGroup(1728,46099)

Overview

Group: SmallGroup(1728,46099)
Rank: 3
Schläfli Type: {3,12}
Vertices, edges, …: 72, 432, 288
Order of s₀s₁s₂: 12
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

3-fold

{3,12}*576

4-fold

12-fold

16-fold

{3,6}*108

36-fold

48-fold

{3,6}*36

72-fold

144-fold

{3,2}*12

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*s2)^2*s1*(s0*(s2*s1)^3)^2> of order 2

144 facets

144 of {3}*6

36 vertex figures

36 of {12}*24

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

144 facets

144 of {3}*6

36 vertex figures

36 of {12}*24

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

144 facets

144 of {3}*6

36 vertex figures

36 of {12}*24

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

144 facets

144 of {3}*6

48 vertex figures

24 of {12}*24
24 of {6}*12

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

144 facets

144 of {3}*6

36 vertex figures

36 of {12}*24

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

96 facets

96 of {3}*6

24 vertex figures

24 of {12}*24

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

96 facets

96 of {3}*6

28 vertex figures

22 of {12}*24
6 of {4}*8

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

72 facets

72 of {3}*6

24 vertex figures

12 of {12}*24
12 of {6}*12

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

72 facets

72 of {3}*6

24 vertex figures

12 of {12}*24
12 of {6}*12

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

72 facets

72 of {3}*6

24 vertex figures

12 of {12}*24
12 of {6}*12

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

72 facets

72 of {3}*6

28 vertex figures

12 of {12}*24
8 of {6}*12
8 of {3}*6

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

72 facets

72 of {3}*6

18 vertex figures

18 of {12}*24

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

48 facets

48 of {3}*6

18 vertex figures

8 of {6}*12
7 of {12}*24
3 of {4}*8

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

48 facets

48 of {3}*6

20 vertex figures

8 of {12}*24
6 of {6}*12
6 of {2}*4

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

48 facets

48 of {3}*6

14 vertex figures

11 of {12}*24
3 of {4}*8

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

48 facets

48 of {3}*6

12 vertex figures

12 of {12}*24

P/N, where N=<s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2, s0*s1*s0*(s2*s1)^4*(s0*s2*s1)^2*s2> of order 6

48 facets

48 of {3}*6

12 vertex figures

12 of {12}*24

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

36 facets

36 of {3}*6

12 vertex figures

6 of {12}*24
6 of {6}*12

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

36 facets

36 of {3}*6

9 vertex figures

9 of {12}*24

P/N, where N=<s0*s1*s0*(s2*s1)^5*s0*s2*s1, s0*s1*(s2*s1*s0)^4*(s2*s1)^2, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^3*s0*(s2*s1)^2> of order 8

36 facets

36 of {3}*6

12 vertex figures

6 of {12}*24
6 of {6}*12

P/N, where N=<s0*s1*s0*(s2*s1)^5*s0*s2*s1, s0*s1*(s2*s1*s0)^4*(s2*s1)^2, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^3*s0*(s2*s1)^2*s2> of order 8

36 facets

36 of {3}*6

12 vertex figures

6 of {12}*24
6 of {6}*12

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

36 facets

36 of {3}*6

12 vertex figures

6 of {12}*24
6 of {6}*12

P/N, where N=<s0*s1*s0*(s2*s1)^5*s0*s2*s1, s0*s1*(s2*s1*s0)^4*(s2*s1)^2, s1*s0*(s2*s1)^2*s0*(s2*s1)^3*s0*(s2*s1)^2*s2> of order 8

36 facets

36 of {3}*6

12 vertex figures

6 of {12}*24
6 of {6}*12

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle