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# Polytope of Type {14,4,4}

Atlas Canonical Name : {14,4,4}*896
Also Known As : {{14,4|2},{4,4}4}. if this polytope has another name.
Group : SmallGroup(896,12517)
Rank : 4
Schlafli Type : {14,4,4}
Number of vertices, edges, etc : 14, 56, 16, 8
Order of s0s1s2s3 : 28
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{14,4,4,2} of size 1792
Vertex Figure Of :
{2,14,4,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {14,4,4}*448
4-fold quotients : {14,2,4}*224, {14,4,2}*224
7-fold quotients : {2,4,4}*128
8-fold quotients : {7,2,4}*112, {14,2,2}*112
14-fold quotients : {2,4,4}*64
16-fold quotients : {7,2,2}*56
28-fold quotients : {2,2,4}*32, {2,4,2}*32
56-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {14,4,8}*1792a, {14,8,4}*1792a, {28,4,4}*1792a, {14,4,4}*1792, {14,4,8}*1792b, {14,8,4}*1792b
Permutation Representation (GAP) :
```s0 := (  1, 29)(  2, 35)(  3, 34)(  4, 33)(  5, 32)(  6, 31)(  7, 30)(  8, 36)
(  9, 42)( 10, 41)( 11, 40)( 12, 39)( 13, 38)( 14, 37)( 15, 43)( 16, 49)
( 17, 48)( 18, 47)( 19, 46)( 20, 45)( 21, 44)( 22, 50)( 23, 56)( 24, 55)
( 25, 54)( 26, 53)( 27, 52)( 28, 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, 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, 51)( 16, 50)
( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 52)( 22, 44)( 23, 43)( 24, 49)
( 25, 48)( 26, 47)( 27, 46)( 28, 45)( 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 := ( 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);;
s3 := (  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)(  8, 64)
(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)
( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)( 24, 80)
( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 85)( 30, 86)( 31, 87)( 32, 88)
( 33, 89)( 34, 90)( 35, 91)( 36, 92)( 37, 93)( 38, 94)( 39, 95)( 40, 96)
( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,101)( 46,102)( 47,103)( 48,104)
( 49,105)( 50,106)( 51,107)( 52,108)( 53,109)( 54,110)( 55,111)( 56,112);;
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(112)!(  1, 29)(  2, 35)(  3, 34)(  4, 33)(  5, 32)(  6, 31)(  7, 30)
(  8, 36)(  9, 42)( 10, 41)( 11, 40)( 12, 39)( 13, 38)( 14, 37)( 15, 43)
( 16, 49)( 17, 48)( 18, 47)( 19, 46)( 20, 45)( 21, 44)( 22, 50)( 23, 56)
( 24, 55)( 25, 54)( 26, 53)( 27, 52)( 28, 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, 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, 51)
( 16, 50)( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 52)( 22, 44)( 23, 43)
( 24, 49)( 25, 48)( 26, 47)( 27, 46)( 28, 45)( 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)!( 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);
s3 := Sym(112)!(  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)
(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)
( 16, 72)( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)
( 24, 80)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 85)( 30, 86)( 31, 87)
( 32, 88)( 33, 89)( 34, 90)( 35, 91)( 36, 92)( 37, 93)( 38, 94)( 39, 95)
( 40, 96)( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,101)( 46,102)( 47,103)
( 48,104)( 49,105)( 50,106)( 51,107)( 52,108)( 53,109)( 54,110)( 55,111)
( 56,112);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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 >;

```
References : None.
to this polytope