Polytope of Type {6,27} with group SmallGroup(1296,1780)

Overview

Group: SmallGroup(1296,1780)
Rank: 3
Schläfli Type: {6,27}
Vertices, edges, …: 24, 324, 108
Order of s₀s₁s₂: 108
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{6,9}*432

4-fold

{6,27}*324

9-fold

{6,3}*144

12-fold

27-fold

{6,3}*48

36-fold

54-fold

{3,3}*24

108-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*s0*s2*(s1*s0)^2*s2*s1> of order 2

54 facets

54 of {6}*12

12 vertex figures

12 of {27}*54

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

54 facets

27 of {2}*4
27 of {6}*12

8 vertex figures

8 of {27}*54

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

27 facets

27 of {6}*12

6 vertex figures

6 of {27}*54

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.