Polytope of Type {4,2,4,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,2,4,27}*1728
if this polytope has a name.
Group : SmallGroup(1728,11358)
Rank : 5
Schlafli Type : {4,2,4,27}
Number of vertices, edges, etc : 4, 4, 4, 54, 27
Order of s₀s₁s₂s₃s₄ : 108
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,4,27}*864
   3-fold quotients : {4,2,4,9}*576
   6-fold quotients : {2,2,4,9}*288
   9-fold quotients : {4,2,4,3}*192
   18-fold quotients : {2,2,4,3}*96
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := (  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)
( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)
( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)
( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)
( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)
( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)
(101,102)(103,104)(105,106)(107,108)(109,110)(111,112);;
s3 := (  6,  7)(  9, 13)( 10, 15)( 11, 14)( 12, 16)( 17, 33)( 18, 35)( 19, 34)
( 20, 36)( 21, 29)( 22, 31)( 23, 30)( 24, 32)( 25, 37)( 26, 39)( 27, 38)
( 28, 40)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 97)( 46, 99)( 47, 98)
( 48,100)( 49, 93)( 50, 95)( 51, 94)( 52, 96)( 53, 77)( 54, 79)( 55, 78)
( 56, 80)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 81)( 62, 83)( 63, 82)
( 64, 84)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)
( 72,104)( 73,109)( 74,111)( 75,110)( 76,112);;
s4 := (  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 49)( 10, 50)( 11, 52)( 12, 51)
( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 69)( 18, 70)( 19, 72)( 20, 71)
( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 73)( 26, 74)( 27, 76)( 28, 75)
( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)( 36, 55)
( 37, 61)( 38, 62)( 39, 64)( 40, 63)( 77, 89)( 78, 90)( 79, 92)( 80, 91)
( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)( 88, 95)
(101,105)(102,106)(103,108)(104,107)(111,112);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(112)!(2,3);
s1 := Sym(112)!(1,2)(3,4);
s2 := Sym(112)!(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)
( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)
( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)
( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)
( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)
( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)
( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112);
s3 := Sym(112)!(  6,  7)(  9, 13)( 10, 15)( 11, 14)( 12, 16)( 17, 33)( 18, 35)
( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)( 24, 32)( 25, 37)( 26, 39)
( 27, 38)( 28, 40)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 97)( 46, 99)
( 47, 98)( 48,100)( 49, 93)( 50, 95)( 51, 94)( 52, 96)( 53, 77)( 54, 79)
( 55, 78)( 56, 80)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 81)( 62, 83)
( 63, 82)( 64, 84)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)
( 71,102)( 72,104)( 73,109)( 74,111)( 75,110)( 76,112);
s4 := Sym(112)!(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 49)( 10, 50)( 11, 52)
( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 69)( 18, 70)( 19, 72)
( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 73)( 26, 74)( 27, 76)
( 28, 75)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)
( 36, 55)( 37, 61)( 38, 62)( 39, 64)( 40, 63)( 77, 89)( 78, 90)( 79, 92)
( 80, 91)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)
( 88, 95)(101,105)(102,106)(103,108)(104,107)(111,112);
poly := sub<Sym(112)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope