Polytope of Type {28,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,4}*896
if this polytope has a name.
Group : SmallGroup(896,1650)
Rank : 3
Schlafli Type : {28,4}
Number of vertices, edges, etc : 112, 224, 16
Order of s₀s₁s₂ : 56
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {28,4,2} of size 1792
Vertex Figure Of :
   {2,28,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,4}*448
   4-fold quotients : {28,4}*224
   7-fold quotients : {4,4}*128
   8-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {4,4}*64
   16-fold quotients : {14,2}*56
   28-fold quotients : {4,4}*32
   32-fold quotients : {7,2}*28
   56-fold quotients : {2,4}*16, {4,2}*16
   112-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {28,8}*1792a, {56,4}*1792a, {28,4}*1792, {28,8}*1792c, {56,4}*1792c
Permutation Representation (GAP) :

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