Polytope of Type {4,56}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,56}*448a
Also Known As : {4,56|2}. if this polytope has another name.
Group : SmallGroup(448,266)
Rank : 3
Schlafli Type : {4,56}
Number of vertices, edges, etc : 4, 112, 56
Order of s₀s₁s₂ : 56
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,56,2} of size 896
   {4,56,4} of size 1792
   {4,56,4} of size 1792
Vertex Figure Of :
   {2,4,56} of size 896
   {4,4,56} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,28}*224, {2,56}*224
   4-fold quotients : {2,28}*112, {4,14}*112
   7-fold quotients : {4,8}*64a
   8-fold quotients : {2,14}*56
   14-fold quotients : {4,4}*32, {2,8}*32
   16-fold quotients : {2,7}*28
   28-fold quotients : {2,4}*16, {4,2}*16
   56-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,56}*896a, {8,56}*896a, {8,56}*896b, {4,112}*896a, {4,112}*896b
   3-fold covers : {12,56}*1344a, {4,168}*1344a
   4-fold covers : {8,56}*1792a, {4,56}*1792a, {8,56}*1792d, {4,112}*1792a, {4,112}*1792b, {16,56}*1792a, {16,56}*1792b, {8,112}*1792c, {8,112}*1792d, {16,56}*1792d, {8,112}*1792e, {8,112}*1792f, {16,56}*1792f, {4,224}*1792a, {4,224}*1792b
Permutation Representation (GAP) :

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