Polytope of Type {108,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {108,2}*432
if this polytope has a name.
Group : SmallGroup(432,45)
Rank : 3
Schlafli Type : {108,2}
Number of vertices, edges, etc : 108, 108, 2
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {108,2,2} of size 864
   {108,2,3} of size 1296
   {108,2,4} of size 1728
Vertex Figure Of :
   {2,108,2} of size 864
   {4,108,2} of size 1728
   {4,108,2} of size 1728
   {4,108,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {54,2}*216
   3-fold quotients : {36,2}*144
   4-fold quotients : {27,2}*108
   6-fold quotients : {18,2}*72
   9-fold quotients : {12,2}*48
   12-fold quotients : {9,2}*36
   18-fold quotients : {6,2}*24
   27-fold quotients : {4,2}*16
   36-fold quotients : {3,2}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {108,4}*864a, {216,2}*864
   3-fold covers : {324,2}*1296, {108,6}*1296a, {108,6}*1296b
   4-fold covers : {216,4}*1728a, {108,4}*1728a, {216,4}*1728b, {108,8}*1728a, {108,8}*1728b, {432,2}*1728, {108,4}*1728b
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 53)
( 44, 52)( 45, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 89)( 59, 88)( 60, 90)
( 61, 86)( 62, 85)( 63, 87)( 64,103)( 65,105)( 66,104)( 67,100)( 68,102)
( 69,101)( 70,107)( 71,106)( 72,108)( 73, 94)( 74, 96)( 75, 95)( 76, 91)
( 77, 93)( 78, 92)( 79, 98)( 80, 97)( 81, 99);;
s1 := (  1, 64)(  2, 66)(  3, 65)(  4, 71)(  5, 70)(  6, 72)(  7, 68)(  8, 67)
(  9, 69)( 10, 55)( 11, 57)( 12, 56)( 13, 62)( 14, 61)( 15, 63)( 16, 59)
( 17, 58)( 18, 60)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)( 24, 74)
( 25, 80)( 26, 79)( 27, 81)( 28, 91)( 29, 93)( 30, 92)( 31, 98)( 32, 97)
( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 82)( 38, 84)( 39, 83)( 40, 89)
( 41, 88)( 42, 90)( 43, 86)( 44, 85)( 45, 87)( 46,103)( 47,105)( 48,104)
( 49,100)( 50,102)( 51,101)( 52,107)( 53,106)( 54,108);;
s2 := (109,110);;
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, 
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*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*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*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*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*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(110)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 22)( 11, 24)( 12, 23)
( 13, 19)( 14, 21)( 15, 20)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 31, 35)
( 32, 34)( 33, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 43, 53)( 44, 52)( 45, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 89)( 59, 88)
( 60, 90)( 61, 86)( 62, 85)( 63, 87)( 64,103)( 65,105)( 66,104)( 67,100)
( 68,102)( 69,101)( 70,107)( 71,106)( 72,108)( 73, 94)( 74, 96)( 75, 95)
( 76, 91)( 77, 93)( 78, 92)( 79, 98)( 80, 97)( 81, 99);
s1 := Sym(110)!(  1, 64)(  2, 66)(  3, 65)(  4, 71)(  5, 70)(  6, 72)(  7, 68)
(  8, 67)(  9, 69)( 10, 55)( 11, 57)( 12, 56)( 13, 62)( 14, 61)( 15, 63)
( 16, 59)( 17, 58)( 18, 60)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)
( 24, 74)( 25, 80)( 26, 79)( 27, 81)( 28, 91)( 29, 93)( 30, 92)( 31, 98)
( 32, 97)( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 82)( 38, 84)( 39, 83)
( 40, 89)( 41, 88)( 42, 90)( 43, 86)( 44, 85)( 45, 87)( 46,103)( 47,105)
( 48,104)( 49,100)( 50,102)( 51,101)( 52,107)( 53,106)( 54,108);
s2 := Sym(110)!(109,110);
poly := sub<Sym(110)|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, 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*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*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*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*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*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 >;

to this polytope