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Polytope of Type {4,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54}*432a
Also Known As : {4,54|2}. if this polytope has another name.
Group : SmallGroup(432,47)
Rank : 3
Schlafli Type : {4,54}
Number of vertices, edges, etc : 4, 108, 54
Order of s0s1s2 : 108
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,54,2} of size 864
   {4,54,4} of size 1728
   {4,54,4} of size 1728
Vertex Figure Of :
   {2,4,54} of size 864
   {4,4,54} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,54}*216
   3-fold quotients : {4,18}*144a
   4-fold quotients : {2,27}*108
   6-fold quotients : {2,18}*72
   9-fold quotients : {4,6}*48a
   12-fold quotients : {2,9}*36
   18-fold quotients : {2,6}*24
   27-fold quotients : {4,2}*16
   36-fold quotients : {2,3}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,108}*864a, {8,54}*864
   3-fold covers : {4,162}*1296a, {12,54}*1296a, {12,54}*1296b
   4-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {16,54}*1728, {4,54}*1728b
Permutation Representation (GAP) :
s0 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)
( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)
( 79,106)( 80,107)( 81,108);;
s1 := (  1, 55)(  2, 57)(  3, 56)(  4, 62)(  5, 61)(  6, 63)(  7, 59)(  8, 58)
(  9, 60)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 80)
( 17, 79)( 18, 81)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)( 24, 65)
( 25, 71)( 26, 70)( 27, 72)( 28, 82)( 29, 84)( 30, 83)( 31, 89)( 32, 88)
( 33, 90)( 34, 86)( 35, 85)( 36, 87)( 37,103)( 38,105)( 39,104)( 40,100)
( 41,102)( 42,101)( 43,107)( 44,106)( 45,108)( 46, 94)( 47, 96)( 48, 95)
( 49, 91)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 99);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 17)(  5, 16)(  6, 18)(  7, 14)(  8, 13)
(  9, 15)( 19, 22)( 20, 24)( 21, 23)( 25, 26)( 28, 37)( 29, 39)( 30, 38)
( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 49)( 47, 51)
( 48, 50)( 52, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)( 60, 72)
( 61, 68)( 62, 67)( 63, 69)( 73, 76)( 74, 78)( 75, 77)( 79, 80)( 82, 91)
( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)
(100,103)(101,105)(102,104)(106,107);;
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*s2*s1*s0*s1*s2*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*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)!( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)
( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)
( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)
( 78,105)( 79,106)( 80,107)( 81,108);
s1 := Sym(108)!(  1, 55)(  2, 57)(  3, 56)(  4, 62)(  5, 61)(  6, 63)(  7, 59)
(  8, 58)(  9, 60)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)
( 16, 80)( 17, 79)( 18, 81)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)
( 24, 65)( 25, 71)( 26, 70)( 27, 72)( 28, 82)( 29, 84)( 30, 83)( 31, 89)
( 32, 88)( 33, 90)( 34, 86)( 35, 85)( 36, 87)( 37,103)( 38,105)( 39,104)
( 40,100)( 41,102)( 42,101)( 43,107)( 44,106)( 45,108)( 46, 94)( 47, 96)
( 48, 95)( 49, 91)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 99);
s2 := Sym(108)!(  1, 10)(  2, 12)(  3, 11)(  4, 17)(  5, 16)(  6, 18)(  7, 14)
(  8, 13)(  9, 15)( 19, 22)( 20, 24)( 21, 23)( 25, 26)( 28, 37)( 29, 39)
( 30, 38)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 49)
( 47, 51)( 48, 50)( 52, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)
( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 76)( 74, 78)( 75, 77)( 79, 80)
( 82, 91)( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)
( 90, 96)(100,103)(101,105)(102,104)(106,107);
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*s2*s1*s0*s1*s2*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*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.
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