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Polytope of Type {56,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,4}*448a
Also Known As : {56,4|2}. if this polytope has another name.
Group : SmallGroup(448,266)
Rank : 3
Schlafli Type : {56,4}
Number of vertices, edges, etc : 56, 112, 4
Order of s0s1s2 : 56
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {56,4,2} of size 896
   {56,4,4} of size 1792
Vertex Figure Of :
   {2,56,4} of size 896
   {4,56,4} of size 1792
   {4,56,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,4}*224, {56,2}*224
   4-fold quotients : {28,2}*112, {14,4}*112
   7-fold quotients : {8,4}*64a
   8-fold quotients : {14,2}*56
   14-fold quotients : {4,4}*32, {8,2}*32
   16-fold quotients : {7,2}*28
   28-fold quotients : {2,4}*16, {4,2}*16
   56-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {56,4}*896a, {56,8}*896a, {56,8}*896b, {112,4}*896a, {112,4}*896b
   3-fold covers : {56,12}*1344a, {168,4}*1344a
   4-fold covers : {56,8}*1792a, {56,4}*1792a, {56,8}*1792d, {112,4}*1792a, {112,4}*1792b, {56,16}*1792a, {56,16}*1792b, {112,8}*1792c, {112,8}*1792d, {56,16}*1792d, {112,8}*1792e, {112,8}*1792f, {56,16}*1792f, {224,4}*1792a, {224,4}*1792b
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, 36)( 30, 42)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 43, 50)( 44, 56)( 45, 55)( 46, 54)( 47, 53)
( 48, 52)( 49, 51)( 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, 99)( 72,105)( 73,104)( 74,103)( 75,102)( 76,101)( 77,100)( 78,106)
( 79,112)( 80,111)( 81,110)( 82,109)( 83,108)( 84,107);;
s1 := (  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)(  8, 65)
(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)( 16, 71)
( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)( 24, 84)
( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 93)( 30, 92)( 31, 98)( 32, 97)
( 33, 96)( 34, 95)( 35, 94)( 36, 86)( 37, 85)( 38, 91)( 39, 90)( 40, 89)
( 41, 88)( 42, 87)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,108)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101);;
s2 := ( 57, 71)( 58, 72)( 59, 73)( 60, 74)( 61, 75)( 62, 76)( 63, 77)( 64, 78)
( 65, 79)( 66, 80)( 67, 81)( 68, 82)( 69, 83)( 70, 84)( 85, 99)( 86,100)
( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)( 94,108)
( 95,109)( 96,110)( 97,111)( 98,112);;
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, 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*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, 36)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 50)( 44, 56)( 45, 55)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 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, 99)( 72,105)( 73,104)( 74,103)( 75,102)( 76,101)( 77,100)
( 78,106)( 79,112)( 80,111)( 81,110)( 82,109)( 83,108)( 84,107);
s1 := Sym(112)!(  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)
(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)
( 16, 71)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)
( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 93)( 30, 92)( 31, 98)
( 32, 97)( 33, 96)( 34, 95)( 35, 94)( 36, 86)( 37, 85)( 38, 91)( 39, 90)
( 40, 89)( 41, 88)( 42, 87)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,108)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)
( 56,101);
s2 := Sym(112)!( 57, 71)( 58, 72)( 59, 73)( 60, 74)( 61, 75)( 62, 76)( 63, 77)
( 64, 78)( 65, 79)( 66, 80)( 67, 81)( 68, 82)( 69, 83)( 70, 84)( 85, 99)
( 86,100)( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)
( 94,108)( 95,109)( 96,110)( 97,111)( 98,112);
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, 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*s0*s1*s0*s1 >; 
 
References : None.
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