Polytope of Type {36,6} with group SmallGroup(1296,1784)

Overview

Group: SmallGroup(1296,1784)
Rank: 3
Schläfli Type: {36,6}
Vertices, edges, …: 108, 324, 18
Order of s₀s₁s₂: 3
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

{12,6}*432d

4-fold

{18,6}*324c

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

References

None.

to this polytope.