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Polytope of Type {8,112}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,112}*1792b
if this polytope has a name.
Group : SmallGroup(1792,82982)
Rank : 3
Schlafli Type : {8,112}
Number of vertices, edges, etc : 8, 448, 112
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}*896d
   4-fold quotients : {4,56}*448b, {8,28}*448a
   7-fold quotients : {8,16}*256b
   8-fold quotients : {4,28}*224, {8,14}*224
   14-fold quotients : {8,8}*128c
   16-fold quotients : {2,28}*112, {4,14}*112
   28-fold quotients : {8,4}*64a, {4,8}*64b
   32-fold quotients : {2,14}*56
   56-fold quotients : {4,4}*32, {8,2}*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 := ( 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);;
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, 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);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*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*s0*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
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, 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);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*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*s0*s1*s2*s1 >; 
 
References : None.
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