Polytope of Type {4,12,6} with group SmallGroup(1728,46671)

Overview

Group: SmallGroup(1728,46671)
Rank: 4
Schläfli Type: {4,12,6}
Vertices, edges, …: 12, 72, 108, 6
Order of s₀s₁s₂s₃: 12
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Orientable
Flat

Quotients maximal quotients in bold

2-fold

{4,6,6}*864k

4-fold

{4,6,3}*432b

9-fold

{4,4,6}*192

18-fold

27-fold

{4,4,2}*64

36-fold

54-fold

72-fold

{2,2,3}*24

108-fold

{2,2,2}*16

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

6 facets

6 of 3-fold non-regular quotient of {4,12}*288

8 vertex figures

2 of {12,6}*144c
6 of {4,6}*48a

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

6 facets

6 of 3-fold non-regular quotient of {4,12}*288

4 vertex figures

4 of {12,6}*144c

Representations

Permutation Representation (GAP)

s0 := (  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)( 48, 51)( 49, 52)( 56, 62)( 57, 60)( 58, 61)( 65, 71)( 66, 69)( 67, 70)( 74, 80)( 75, 78)( 76, 79)( 83, 89)( 84, 87)( 85, 88)( 92, 98)( 93, 96)( 94, 97)(101,107)(102,105)(103,106);;
s1 := (  4,  9)(  5,  7)(  6,  8)( 13, 18)( 14, 16)( 15, 17)( 22, 27)( 23, 25)( 24, 26)( 31, 36)( 32, 34)( 33, 35)( 40, 45)( 41, 43)( 42, 44)( 49, 54)( 50, 52)( 51, 53)( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)( 60, 89)( 61, 86)( 62, 87)( 63, 85)( 64, 91)( 65, 92)( 66, 93)( 67, 99)( 68, 97)( 69, 98)( 70, 95)( 71, 96)( 72, 94)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)( 79,104)( 80,105)( 81,103);;
s2 := (  1, 59)(  2, 58)(  3, 60)(  4, 56)(  5, 55)(  6, 57)(  7, 62)(  8, 61)(  9, 63)( 10, 77)( 11, 76)( 12, 78)( 13, 74)( 14, 73)( 15, 75)( 16, 80)( 17, 79)( 18, 81)( 19, 68)( 20, 67)( 21, 69)( 22, 65)( 23, 64)( 24, 66)( 25, 71)( 26, 70)( 27, 72)( 28, 86)( 29, 85)( 30, 87)( 31, 83)( 32, 82)( 33, 84)( 34, 89)( 35, 88)( 36, 90)( 37,104)( 38,103)( 39,105)( 40,101)( 41,100)( 42,102)( 43,107)( 44,106)( 45,108)( 46, 95)( 47, 94)( 48, 96)( 49, 92)( 50, 91)( 51, 93)( 52, 98)( 53, 97)( 54, 99);;
s3 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)(101,102)(103,106)(104,108)(105,107);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s3*s2*s3*s2*s1*s0*s1*s2*s3*s2*s3*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(108)!(  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)( 48, 51)( 49, 52)( 56, 62)( 57, 60)( 58, 61)( 65, 71)( 66, 69)( 67, 70)( 74, 80)( 75, 78)( 76, 79)( 83, 89)( 84, 87)( 85, 88)( 92, 98)( 93, 96)( 94, 97)(101,107)(102,105)(103,106);
s1 := Sym(108)!(  4,  9)(  5,  7)(  6,  8)( 13, 18)( 14, 16)( 15, 17)( 22, 27)( 23, 25)( 24, 26)( 31, 36)( 32, 34)( 33, 35)( 40, 45)( 41, 43)( 42, 44)( 49, 54)( 50, 52)( 51, 53)( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)( 60, 89)( 61, 86)( 62, 87)( 63, 85)( 64, 91)( 65, 92)( 66, 93)( 67, 99)( 68, 97)( 69, 98)( 70, 95)( 71, 96)( 72, 94)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)( 79,104)( 80,105)( 81,103);
s2 := Sym(108)!(  1, 59)(  2, 58)(  3, 60)(  4, 56)(  5, 55)(  6, 57)(  7, 62)(  8, 61)(  9, 63)( 10, 77)( 11, 76)( 12, 78)( 13, 74)( 14, 73)( 15, 75)( 16, 80)( 17, 79)( 18, 81)( 19, 68)( 20, 67)( 21, 69)( 22, 65)( 23, 64)( 24, 66)( 25, 71)( 26, 70)( 27, 72)( 28, 86)( 29, 85)( 30, 87)( 31, 83)( 32, 82)( 33, 84)( 34, 89)( 35, 88)( 36, 90)( 37,104)( 38,103)( 39,105)( 40,101)( 41,100)( 42,102)( 43,107)( 44,106)( 45,108)( 46, 95)( 47, 94)( 48, 96)( 49, 92)( 50, 91)( 51, 93)( 52, 98)( 53, 97)( 54, 99);
s3 := Sym(108)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)(101,102)(103,106)(104,108)(105,107);
poly := sub<Sym(108)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s3*s2*s3*s2*s1*s0*s1*s2*s3*s2*s3*s2*s1 >;

References

None.

to this polytope.