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Polytope of Type {4,4,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,14}*896
Also Known As : {{4,4}4,{4,14|2}}. if this polytope has another name.
Group : SmallGroup(896,12517)
Rank : 4
Schlafli Type : {4,4,14}
Number of vertices, edges, etc : 8, 16, 56, 14
Order of s0s1s2s3 : 28
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,14,2} of size 1792
Vertex Figure Of :
   {2,4,4,14} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4,14}*448
   4-fold quotients : {2,4,14}*224, {4,2,14}*224
   7-fold quotients : {4,4,2}*128
   8-fold quotients : {4,2,7}*112, {2,2,14}*112
   14-fold quotients : {4,4,2}*64
   16-fold quotients : {2,2,7}*56
   28-fold quotients : {2,4,2}*32, {4,2,2}*32
   56-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8,14}*1792a, {8,4,14}*1792a, {4,4,28}*1792a, {4,4,14}*1792, {4,8,14}*1792b, {8,4,14}*1792b
Permutation Representation (GAP) :
s0 := ( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 99,106)(100,107)(101,108)
(102,109)(103,110)(104,111)(105,112);;
s1 := ( 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);;
s2 := (  1, 85)(  2, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)(  8, 92)
(  9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 93)( 15, 99)( 16,105)
( 17,104)( 18,103)( 19,102)( 20,101)( 21,100)( 22,106)( 23,112)( 24,111)
( 25,110)( 26,109)( 27,108)( 28,107)( 29, 57)( 30, 63)( 31, 62)( 32, 61)
( 33, 60)( 34, 59)( 35, 58)( 36, 64)( 37, 70)( 38, 69)( 39, 68)( 40, 67)
( 41, 66)( 42, 65)( 43, 71)( 44, 77)( 45, 76)( 46, 75)( 47, 74)( 48, 73)
( 49, 72)( 50, 78)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 79);;
s3 := (  1, 30)(  2, 29)(  3, 35)(  4, 34)(  5, 33)(  6, 32)(  7, 31)(  8, 37)
(  9, 36)( 10, 42)( 11, 41)( 12, 40)( 13, 39)( 14, 38)( 15, 44)( 16, 43)
( 17, 49)( 18, 48)( 19, 47)( 20, 46)( 21, 45)( 22, 51)( 23, 50)( 24, 56)
( 25, 55)( 26, 54)( 27, 53)( 28, 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,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(112)!( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)
( 43, 50)( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 71, 78)
( 72, 79)( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 99,106)(100,107)
(101,108)(102,109)(103,110)(104,111)(105,112);
s1 := 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);
s2 := Sym(112)!(  1, 85)(  2, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)
(  8, 92)(  9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 93)( 15, 99)
( 16,105)( 17,104)( 18,103)( 19,102)( 20,101)( 21,100)( 22,106)( 23,112)
( 24,111)( 25,110)( 26,109)( 27,108)( 28,107)( 29, 57)( 30, 63)( 31, 62)
( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 64)( 37, 70)( 38, 69)( 39, 68)
( 40, 67)( 41, 66)( 42, 65)( 43, 71)( 44, 77)( 45, 76)( 46, 75)( 47, 74)
( 48, 73)( 49, 72)( 50, 78)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)
( 56, 79);
s3 := Sym(112)!(  1, 30)(  2, 29)(  3, 35)(  4, 34)(  5, 33)(  6, 32)(  7, 31)
(  8, 37)(  9, 36)( 10, 42)( 11, 41)( 12, 40)( 13, 39)( 14, 38)( 15, 44)
( 16, 43)( 17, 49)( 18, 48)( 19, 47)( 20, 46)( 21, 45)( 22, 51)( 23, 50)
( 24, 56)( 25, 55)( 26, 54)( 27, 53)( 28, 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,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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