Polytope of Type {36,6} with group SmallGroup(1296,1785)

Overview

Group: SmallGroup(1296,1785)
Rank: 3
Schläfli Type: {36,6}
Vertices, edges, …: 108, 324, 18
Order of s₀s₁s₂: 9
Order of s₀s₁s₂s₁: 12
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

3-fold

4-fold

{18,6}*324a

9-fold

{12,6}*144d

12-fold

{6,6}*108

27-fold

{4,6}*48b

54-fold

{4,3}*24

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

12 facets

6 of {18}*36
6 of {36}*72

54 vertex figures

54 of {6}*12

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

6 facets

6 of {36}*72

36 vertex figures

36 of {6}*12

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.