Polytope of Type {112,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {112,8}*1792a
if this polytope has a name.
Group : SmallGroup(1792,82962)
Rank : 3
Schlafli Type : {112,8}
Number of vertices, edges, etc : 112, 448, 8
Order of s₀s₁s₂ : 112
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {56,8}*896d
   4-fold quotients : {56,4}*448b, {28,8}*448a
   7-fold quotients : {16,8}*256a
   8-fold quotients : {28,4}*224, {14,8}*224
   14-fold quotients : {8,8}*128a
   16-fold quotients : {28,2}*112, {14,4}*112
   28-fold quotients : {4,8}*64a, {8,4}*64b
   32-fold quotients : {14,2}*56
   56-fold quotients : {4,4}*32, {2,8}*32
   64-fold quotients : {7,2}*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, 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, 85)( 30, 91)( 31, 90)( 32, 89)
( 33, 88)( 34, 87)( 35, 86)( 36, 92)( 37, 98)( 38, 97)( 39, 96)( 40, 95)
( 41, 94)( 42, 93)( 43, 99)( 44,105)( 45,104)( 46,103)( 47,102)( 48,101)
( 49,100)( 50,106)( 51,112)( 52,111)( 53,110)( 54,109)( 55,108)( 56,107);;
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, 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, 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 := ( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 29, 43)
( 30, 44)( 31, 45)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 50)( 37, 51)
( 38, 52)( 39, 53)( 40, 54)( 41, 55)( 42, 56)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := 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, 85)( 30, 91)( 31, 90)
( 32, 89)( 33, 88)( 34, 87)( 35, 86)( 36, 92)( 37, 98)( 38, 97)( 39, 96)
( 40, 95)( 41, 94)( 42, 93)( 43, 99)( 44,105)( 45,104)( 46,103)( 47,102)
( 48,101)( 49,100)( 50,106)( 51,112)( 52,111)( 53,110)( 54,109)( 55,108)
( 56,107);
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, 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, 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)!( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)
( 29, 43)( 30, 44)( 31, 45)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 50)
( 37, 51)( 38, 52)( 39, 53)( 40, 54)( 41, 55)( 42, 56)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >;

References : None.
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