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Polytope of Type {16,56}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,56}*1792b
if this polytope has a name.
Group : SmallGroup(1792,82983)
Rank : 3
Schlafli Type : {16,56}
Number of vertices, edges, etc : 16, 448, 56
Order of s0s1s2 : 112
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {8,56}*896a
   4-fold quotients : {4,56}*448a, {8,28}*448b
   7-fold quotients : {16,8}*256b
   8-fold quotients : {4,28}*224, {2,56}*224
   14-fold quotients : {8,8}*128a
   16-fold quotients : {2,28}*112, {4,14}*112
   28-fold quotients : {4,8}*64a, {8,4}*64b
   32-fold quotients : {2,14}*56
   56-fold quotients : {4,4}*32, {2,8}*32
   64-fold quotients : {2,7}*28
   112-fold quotients : {2,4}*16, {4,2}*16
   224-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  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, 78)( 16, 79)
( 17, 80)( 18, 81)( 19, 82)( 20, 83)( 21, 84)( 22, 71)( 23, 72)( 24, 73)
( 25, 74)( 26, 75)( 27, 76)( 28, 77)( 29,106)( 30,107)( 31,108)( 32,109)
( 33,110)( 34,111)( 35,112)( 36, 99)( 37,100)( 38,101)( 39,102)( 40,103)
( 41,104)( 42,105)( 43, 92)( 44, 93)( 45, 94)( 46, 95)( 47, 96)( 48, 97)
( 49, 98)( 50, 85)( 51, 86)( 52, 87)( 53, 88)( 54, 89)( 55, 90)( 56, 91);;
s1 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 15, 22)( 16, 28)
( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 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)( 57, 85)( 58, 91)( 59, 90)( 60, 89)( 61, 88)
( 62, 87)( 63, 86)( 64, 92)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)
( 70, 93)( 71,106)( 72,112)( 73,111)( 74,110)( 75,109)( 76,108)( 77,107)
( 78, 99)( 79,105)( 80,104)( 81,103)( 82,102)( 83,101)( 84,100);;
s2 := (  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, 44)( 30, 43)( 31, 49)
( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 51)( 37, 50)( 38, 56)( 39, 55)
( 40, 54)( 41, 53)( 42, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)
( 67, 69)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)( 76, 81)( 77, 80)
( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)( 92,107)
( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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, 78)
( 16, 79)( 17, 80)( 18, 81)( 19, 82)( 20, 83)( 21, 84)( 22, 71)( 23, 72)
( 24, 73)( 25, 74)( 26, 75)( 27, 76)( 28, 77)( 29,106)( 30,107)( 31,108)
( 32,109)( 33,110)( 34,111)( 35,112)( 36, 99)( 37,100)( 38,101)( 39,102)
( 40,103)( 41,104)( 42,105)( 43, 92)( 44, 93)( 45, 94)( 46, 95)( 47, 96)
( 48, 97)( 49, 98)( 50, 85)( 51, 86)( 52, 87)( 53, 88)( 54, 89)( 55, 90)
( 56, 91);
s1 := Sym(112)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 15, 22)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 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)( 57, 85)( 58, 91)( 59, 90)( 60, 89)
( 61, 88)( 62, 87)( 63, 86)( 64, 92)( 65, 98)( 66, 97)( 67, 96)( 68, 95)
( 69, 94)( 70, 93)( 71,106)( 72,112)( 73,111)( 74,110)( 75,109)( 76,108)
( 77,107)( 78, 99)( 79,105)( 80,104)( 81,103)( 82,102)( 83,101)( 84,100);
s2 := 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, 44)( 30, 43)
( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 51)( 37, 50)( 38, 56)
( 39, 55)( 40, 54)( 41, 53)( 42, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 65)
( 66, 70)( 67, 69)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)( 76, 81)
( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)
( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2 >; 
 
References : None.
to this polytope