# Polytope of Type {16,14}

Atlas Canonical Name : {16,14}*448
Also Known As : {16,14|2}. if this polytope has another name.
Group : SmallGroup(448,444)
Rank : 3
Schlafli Type : {16,14}
Number of vertices, edges, etc : 16, 112, 14
Order of s0s1s2 : 112
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{16,14,2} of size 896
{16,14,4} of size 1792
Vertex Figure Of :
{2,16,14} of size 896
{4,16,14} of size 1792
{4,16,14} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,14}*224
4-fold quotients : {4,14}*112
7-fold quotients : {16,2}*64
8-fold quotients : {2,14}*56
14-fold quotients : {8,2}*32
16-fold quotients : {2,7}*28
28-fold quotients : {4,2}*16
56-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {16,28}*896a, {32,14}*896
3-fold covers : {48,14}*1344, {16,42}*1344
4-fold covers : {16,28}*1792a, {16,56}*1792c, {16,56}*1792d, {32,28}*1792a, {32,28}*1792b, {64,14}*1792
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)( 57, 85)( 58, 86)( 59, 87)
( 60, 88)( 61, 89)( 62, 90)( 63, 91)( 64, 92)( 65, 93)( 66, 94)( 67, 95)
( 68, 96)( 69, 97)( 70, 98)( 71,106)( 72,107)( 73,108)( 74,109)( 75,110)
( 76,111)( 77,112)( 78, 99)( 79,100)( 80,101)( 81,102)( 82,103)( 83,104)
( 84,105);;
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, 78)( 16, 84)
( 17, 83)( 18, 82)( 19, 81)( 20, 80)( 21, 79)( 22, 71)( 23, 77)( 24, 76)
( 25, 75)( 26, 74)( 27, 73)( 28, 72)( 29, 99)( 30,105)( 31,104)( 32,103)
( 33,102)( 34,101)( 35,100)( 36,106)( 37,112)( 38,111)( 39,110)( 40,109)
( 41,108)( 42,107)( 43, 85)( 44, 91)( 45, 90)( 46, 89)( 47, 88)( 48, 87)
( 49, 86)( 50, 92)( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)( 56, 93);;
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, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)
( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)(109,111);;
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*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,
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*s0*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)( 57, 85)( 58, 86)
( 59, 87)( 60, 88)( 61, 89)( 62, 90)( 63, 91)( 64, 92)( 65, 93)( 66, 94)
( 67, 95)( 68, 96)( 69, 97)( 70, 98)( 71,106)( 72,107)( 73,108)( 74,109)
( 75,110)( 76,111)( 77,112)( 78, 99)( 79,100)( 80,101)( 81,102)( 82,103)
( 83,104)( 84,105);
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, 78)
( 16, 84)( 17, 83)( 18, 82)( 19, 81)( 20, 80)( 21, 79)( 22, 71)( 23, 77)
( 24, 76)( 25, 75)( 26, 74)( 27, 73)( 28, 72)( 29, 99)( 30,105)( 31,104)
( 32,103)( 33,102)( 34,101)( 35,100)( 36,106)( 37,112)( 38,111)( 39,110)
( 40,109)( 41,108)( 42,107)( 43, 85)( 44, 91)( 45, 90)( 46, 89)( 47, 88)
( 48, 87)( 49, 86)( 50, 92)( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)
( 56, 93);
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, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)
( 53, 55)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)
( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)
(109,111);
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*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,
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*s0*s1 >;

```
References : None.
to this polytope