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

Overview

Group: SmallGroup(1728,46099)
Rank: 3
Schläfli Type: {12,3}
Vertices, edges, …: 288, 432, 72
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
Self-Petrie

Quotients maximal quotients in bold

3-fold

{12,3}*576

4-fold

12-fold

16-fold

{6,3}*108

36-fold

48-fold

{6,3}*36

72-fold

144-fold

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

36 facets

36 of {12}*24

144 vertex figures

144 of {3}*6

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

36 facets

36 of {12}*24

144 vertex figures

144 of {3}*6

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

36 facets

36 of {12}*24

144 vertex figures

144 of {3}*6

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

48 facets

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

144 vertex figures

144 of {3}*6

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

36 facets

36 of {12}*24

144 vertex figures

144 of {3}*6

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

24 facets

24 of {12}*24

96 vertex figures

96 of {3}*6

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

28 facets

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

96 vertex figures

96 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

P/N, where N=<(s0*s1)^6, (s0*s1)^3*s2*(s1*s0)^3*s2*(s1*s0)^2*s1*s2> of order 4

24 facets

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

72 vertex figures

72 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

P/N, where N=<(s0*s1)^6, (s0*s1)^2*s2*(s1*s0)^5*s1*s2*s1> of order 4

24 facets

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

72 vertex figures

72 of {3}*6

P/N, where N=<(s0*s1)^6, ((s1*s0)^2*s1*s2)^3> of order 4

24 facets

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

72 vertex figures

72 of {3}*6

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

28 facets

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

72 vertex figures

72 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

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

18 facets

18 of {12}*24

72 vertex figures

72 of {3}*6

P/N, where N=<(s0*s1)^4, s1*s0*s1*s2*(s1*s0)^5*s1*s2> of order 6

18 facets

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

48 vertex figures

48 of {3}*6

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

20 facets

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

48 vertex figures

48 of {3}*6

P/N, where N=<(s0*s1)^4, (s0*s1)^2*s2*(s1*s0)^5*s1*s2> of order 6

14 facets

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

48 vertex figures

48 of {3}*6

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

12 facets

12 of {12}*24

48 vertex figures

48 of {3}*6

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^3*s2, (s0*s1)^2*s2*(s1*s0)^5*s1*s2*s1> of order 6

12 facets

12 of {12}*24

48 vertex figures

48 of {3}*6

P/N, where N=<(s0*s1)^6, ((s1*s0)^2*s1*s2)^3, (s1*s0)^2*s2*(s1*s0)^3*s2*(s1*s0)^2*s1*s2*s1> of order 8

12 facets

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

36 vertex figures

36 of {3}*6

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

9 facets

9 of {12}*24

36 vertex figures

36 of {3}*6

P/N, where N=<(s0*s1)^6, s1*s0*(s1*s2*(s1*s0)^2)^2*s1*s2*s1> of order 8

12 facets

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

36 vertex figures

36 of {3}*6

P/N, where N=<(s0*s1)^6, ((s1*s0)^2*s1*s2)^3, (s0*s1)^2*s0*s2*(s1*s0)^3*s2*(s1*s0)^2*s1*s2*s1> of order 8

12 facets

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

36 vertex figures

36 of {3}*6

P/N, where N=<(s0*s1)^6, s0*s1*s0*(s2*(s1*s0)^2*s1)^2*s2*s1> of order 8

12 facets

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

36 vertex figures

36 of {3}*6

P/N, where N=<(s0*s1)^6, s0*s1*s2*(s1*s0)^3*s2*(s1*s0)^2*s1*s2*s1> of order 8

12 facets

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

36 vertex figures

36 of {3}*6

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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