Polytope of Type {18,12} with group SmallGroup(432,315)

Overview

Group: SmallGroup(432,315)
Rank: 3
Schläfli Type: {18,12}
Vertices, edges, …: 18, 108, 12
Order of s₀s₁s₂: 36
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{18,6}*216b

3-fold

4-fold

{9,6}*108

6-fold

9-fold

{6,4}*48a

12-fold

18-fold

{6,2}*24

27-fold

{2,4}*16

36-fold

{3,2}*12

54-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References