Polytope of Type {54,4}

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

References : None.
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