Polytope of Type {2,54,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,54,4}*864a
if this polytope has a name.
Group : SmallGroup(864,635)
Rank : 4
Schlafli Type : {2,54,4}
Number of vertices, edges, etc : 2, 54, 108, 4
Order of s₀s₁s₂s₃ : 108
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,54,4,2} of size 1728
Vertex Figure Of :
   {2,2,54,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,54,2}*432
   3-fold quotients : {2,18,4}*288a
   4-fold quotients : {2,27,2}*216
   6-fold quotients : {2,18,2}*144
   9-fold quotients : {2,6,4}*96a
   12-fold quotients : {2,9,2}*72
   18-fold quotients : {2,6,2}*48
   27-fold quotients : {2,2,4}*32
   36-fold quotients : {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,108,4}*1728a, {4,54,4}*1728a, {2,54,8}*1728
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6, 10)(  7,  9)(  8, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 21)
( 16, 23)( 17, 22)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 33, 37)( 34, 36)
( 35, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)( 45, 55)
( 46, 54)( 47, 56)( 58, 59)( 60, 64)( 61, 63)( 62, 65)( 66, 78)( 67, 80)
( 68, 79)( 69, 75)( 70, 77)( 71, 76)( 72, 82)( 73, 81)( 74, 83)( 85, 86)
( 87, 91)( 88, 90)( 89, 92)( 93,105)( 94,107)( 95,106)( 96,102)( 97,104)
( 98,103)( 99,109)(100,108)(101,110);;
s2 := (  3, 12)(  4, 14)(  5, 13)(  6, 19)(  7, 18)(  8, 20)(  9, 16)( 10, 15)
( 11, 17)( 21, 24)( 22, 26)( 23, 25)( 27, 28)( 30, 39)( 31, 41)( 32, 40)
( 33, 46)( 34, 45)( 35, 47)( 36, 43)( 37, 42)( 38, 44)( 48, 51)( 49, 53)
( 50, 52)( 54, 55)( 57, 93)( 58, 95)( 59, 94)( 60,100)( 61, 99)( 62,101)
( 63, 97)( 64, 96)( 65, 98)( 66, 84)( 67, 86)( 68, 85)( 69, 91)( 70, 90)
( 71, 92)( 72, 88)( 73, 87)( 74, 89)( 75,105)( 76,107)( 77,106)( 78,102)
( 79,104)( 80,103)( 81,109)( 82,108)( 83,110);;
s3 := (  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)( 55,109)( 56,110);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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(110)!(1,2);
s1 := Sym(110)!(  4,  5)(  6, 10)(  7,  9)(  8, 11)( 12, 24)( 13, 26)( 14, 25)
( 15, 21)( 16, 23)( 17, 22)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 33, 37)
( 34, 36)( 35, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)
( 45, 55)( 46, 54)( 47, 56)( 58, 59)( 60, 64)( 61, 63)( 62, 65)( 66, 78)
( 67, 80)( 68, 79)( 69, 75)( 70, 77)( 71, 76)( 72, 82)( 73, 81)( 74, 83)
( 85, 86)( 87, 91)( 88, 90)( 89, 92)( 93,105)( 94,107)( 95,106)( 96,102)
( 97,104)( 98,103)( 99,109)(100,108)(101,110);
s2 := Sym(110)!(  3, 12)(  4, 14)(  5, 13)(  6, 19)(  7, 18)(  8, 20)(  9, 16)
( 10, 15)( 11, 17)( 21, 24)( 22, 26)( 23, 25)( 27, 28)( 30, 39)( 31, 41)
( 32, 40)( 33, 46)( 34, 45)( 35, 47)( 36, 43)( 37, 42)( 38, 44)( 48, 51)
( 49, 53)( 50, 52)( 54, 55)( 57, 93)( 58, 95)( 59, 94)( 60,100)( 61, 99)
( 62,101)( 63, 97)( 64, 96)( 65, 98)( 66, 84)( 67, 86)( 68, 85)( 69, 91)
( 70, 90)( 71, 92)( 72, 88)( 73, 87)( 74, 89)( 75,105)( 76,107)( 77,106)
( 78,102)( 79,104)( 80,103)( 81,109)( 82,108)( 83,110);
s3 := Sym(110)!(  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)( 55,109)( 56,110);
poly := sub<Sym(110)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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 >;

to this polytope