Polytope of Type {4,4} with group SmallGroup(1152,32554)

Overview

Group: SmallGroup(1152,32554)
Rank: 3
Schläfli Type: {4,4}
Vertices, edges, …: 144, 288, 144
Order of s₀s₁s₂: 24
Order of s₀s₁s₂s₁: 12
Also known as: {4,4}_(12,0), {4,4|12}. if this polytope has another name.

Special Properties

Toroidal
Locally Spherical
Orientable
Self-Dual

Quotients maximal quotients in bold

2-fold

{4,4}*576

4-fold

{4,4}*288

8-fold

{4,4}*144

9-fold

{4,4}*128

16-fold

{4,4}*72

18-fold

{4,4}*64

36-fold

{4,4}*32

72-fold

144-fold

{2,2}*8

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*s1)^3*s0*s1*(s2*s1*s0)^4*s2*s1*s2> of order 2

72 facets

72 of {4}*8

72 vertex figures

72 of {4}*8

P/N, where N=<(s1*s0*s2)^11*s1*s2> of order 2

72 facets

72 of {4}*8

72 vertex figures

72 of {4}*8

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

72 facets

72 of {4}*8

74 vertex figures

70 of {4}*8
4 of {2}*4

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

74 facets

4 of {2}*4
70 of {4}*8

72 vertex figures

72 of {4}*8

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

72 facets

72 of {4}*8

72 vertex figures

72 of {4}*8

P/N, where N=<s0*(s1*s0*s2)^7*s1*s2> of order 3

48 facets

48 of {4}*8

48 vertex figures

48 of {4}*8

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

48 facets

48 of {4}*8

48 vertex figures

48 of {4}*8

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

36 facets

36 of {4}*8

38 vertex figures

34 of {4}*8
4 of {2}*4

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

36 facets

36 of {4}*8

36 vertex figures

36 of {4}*8

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

36 facets

36 of {4}*8

36 vertex figures

36 of {4}*8

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

36 facets

36 of {4}*8

36 vertex figures

36 of {4}*8

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

36 facets

36 of {4}*8

36 vertex figures

36 of {4}*8

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

36 facets

36 of {4}*8

36 vertex figures

36 of {4}*8

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

36 facets

36 of {4}*8

37 vertex figures

35 of {4}*8
2 of {2}*4

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

38 facets

4 of {2}*4
34 of {4}*8

36 vertex figures

36 of {4}*8

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^10*s2*s1*s2> of order 4

37 facets

2 of {2}*4
35 of {4}*8

36 vertex figures

36 of {4}*8

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

36 facets

36 of {4}*8

36 vertex figures

36 of {4}*8

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

26 vertex figures

22 of {4}*8
4 of {2}*4

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

26 facets

4 of {2}*4
22 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

26 vertex figures

4 of {2}*4
22 of {4}*8

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

26 facets

4 of {2}*4
22 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

24 facets

24 of {4}*8

24 vertex figures

24 of {4}*8

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

18 facets

18 of {4}*8

18 vertex figures

18 of {4}*8

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

18 facets

18 of {4}*8

19 vertex figures

17 of {4}*8
2 of {2}*4

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

19 facets

2 of {2}*4
17 of {4}*8

18 vertex figures

18 of {4}*8

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

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

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

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1*s0*s1)^4*s2*s1*s0*(s1*s2)^2> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

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

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

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

12 facets

12 of {4}*8

14 vertex figures

10 of {4}*8
4 of {2}*4

P/N, where N=<(s1*s2*s1*s0)^2*(s1*s2)^2, (s0*s1*s2*s1)^3*s0*s2, (s1*s0*s1*s2)^4> of order 12

12 facets

12 of {4}*8

14 vertex figures

10 of {4}*8
4 of {2}*4

P/N, where N=<(s1*s2)^2, s0*s1*(s2*s1*s0)^5*s2> of order 12

12 facets

12 of {4}*8

14 vertex figures

4 of {2}*4
10 of {4}*8

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1*s2*s1)^5*s0*(s1*s2)^2> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2*s0*(s1*s2)^2, (s1*s0*s1*s2)^4> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<s0*s1*s0*(s2*s1*s0*s1)^2*s2*s1, (s1*s0*s1*s2)^4> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

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

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^3, (s1*s0*s1*s2)^4> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1)^2, s0*(s2*s1*s0*s1)^3*s2, (s0*s1*s2*s1)^6> of order 12

14 facets

4 of {2}*4
10 of {4}*8

12 vertex figures

12 of {4}*8

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

14 facets

4 of {2}*4
10 of {4}*8

12 vertex figures

12 of {4}*8

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

12 facets

12 of {4}*8

13 vertex figures

11 of {4}*8
2 of {2}*4

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1*s0*s1)^5*s2*s1> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2, (s0*s1*s2*s1)^6> of order 12

12 facets

12 of {4}*8

12 vertex figures

12 of {4}*8

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

14 facets

4 of {2}*4
10 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1)^2, s0*(s2*s1*s0*s1)^3*s2, (s0*s1*s2*s1)^4*s0*s1*(s2*s1*s0)^2*s2*s1*s2> of order 12

13 facets

2 of {2}*4
11 of {4}*8

12 vertex figures

12 of {4}*8

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2, (s0*s1*s2*s1)^3> of order 24

6 facets

6 of {4}*8

6 vertex figures

6 of {4}*8

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

6 facets

6 of {4}*8

7 vertex figures

5 of {4}*8
2 of {2}*4

P/N, where N=<(s0*s1)^2, s0*(s2*s1*s0*s1)^3*s2, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s2*s1*s2> of order 24

7 facets

2 of {2}*4
5 of {4}*8

6 vertex figures

6 of {4}*8

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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