# Polytope of Type {14,2,2}

Atlas Canonical Name : {14,2,2}*112
if this polytope has a name.
Group : SmallGroup(112,42)
Rank : 4
Schlafli Type : {14,2,2}
Number of vertices, edges, etc : 14, 14, 2, 2
Order of s0s1s2s3 : 14
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{14,2,2,2} of size 224
{14,2,2,3} of size 336
{14,2,2,4} of size 448
{14,2,2,5} of size 560
{14,2,2,6} of size 672
{14,2,2,7} of size 784
{14,2,2,8} of size 896
{14,2,2,9} of size 1008
{14,2,2,10} of size 1120
{14,2,2,11} of size 1232
{14,2,2,12} of size 1344
{14,2,2,13} of size 1456
{14,2,2,14} of size 1568
{14,2,2,15} of size 1680
{14,2,2,16} of size 1792
{14,2,2,17} of size 1904
Vertex Figure Of :
{2,14,2,2} of size 224
{4,14,2,2} of size 448
{6,14,2,2} of size 672
{7,14,2,2} of size 784
{8,14,2,2} of size 896
{10,14,2,2} of size 1120
{12,14,2,2} of size 1344
{4,14,2,2} of size 1568
{14,14,2,2} of size 1568
{14,14,2,2} of size 1568
{14,14,2,2} of size 1568
{16,14,2,2} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {7,2,2}*56
7-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {28,2,2}*224, {14,2,4}*224, {14,4,2}*224
3-fold covers : {14,2,6}*336, {14,6,2}*336, {42,2,2}*336
4-fold covers : {28,4,2}*448, {28,2,4}*448, {14,4,4}*448, {56,2,2}*448, {14,2,8}*448, {14,8,2}*448
5-fold covers : {14,2,10}*560, {14,10,2}*560, {70,2,2}*560
6-fold covers : {14,2,12}*672, {14,12,2}*672, {28,2,6}*672, {28,6,2}*672a, {14,4,6}*672, {14,6,4}*672a, {84,2,2}*672, {42,2,4}*672, {42,4,2}*672a
7-fold covers : {98,2,2}*784, {14,2,14}*784, {14,14,2}*784a, {14,14,2}*784c
8-fold covers : {28,4,4}*896, {56,4,2}*896a, {28,4,2}*896, {56,4,2}*896b, {28,8,2}*896a, {28,8,2}*896b, {56,2,4}*896, {28,2,8}*896, {14,4,8}*896a, {14,8,4}*896a, {14,4,8}*896b, {14,8,4}*896b, {14,4,4}*896, {112,2,2}*896, {14,2,16}*896, {14,16,2}*896
9-fold covers : {14,2,18}*1008, {14,18,2}*1008, {126,2,2}*1008, {14,6,6}*1008a, {14,6,6}*1008b, {14,6,6}*1008c, {42,6,2}*1008a, {42,2,6}*1008, {42,6,2}*1008b, {42,6,2}*1008c
10-fold covers : {14,2,20}*1120, {14,20,2}*1120, {28,2,10}*1120, {28,10,2}*1120, {14,4,10}*1120, {14,10,4}*1120, {140,2,2}*1120, {70,2,4}*1120, {70,4,2}*1120
11-fold covers : {14,2,22}*1232, {14,22,2}*1232, {154,2,2}*1232
12-fold covers : {28,2,12}*1344, {28,6,4}*1344a, {14,4,12}*1344, {14,12,4}*1344a, {28,4,6}*1344, {14,2,24}*1344, {14,24,2}*1344, {56,2,6}*1344, {56,6,2}*1344, {14,6,8}*1344, {14,8,6}*1344, {28,12,2}*1344, {84,4,2}*1344a, {84,2,4}*1344, {42,4,4}*1344, {168,2,2}*1344, {42,2,8}*1344, {42,8,2}*1344, {14,4,6}*1344, {14,6,4}*1344, {14,6,6}*1344, {28,6,2}*1344, {42,6,2}*1344, {42,4,2}*1344
13-fold covers : {14,2,26}*1456, {14,26,2}*1456, {182,2,2}*1456
14-fold covers : {196,2,2}*1568, {98,2,4}*1568, {98,4,2}*1568, {14,2,28}*1568, {14,28,2}*1568a, {28,2,14}*1568, {28,14,2}*1568a, {28,14,2}*1568b, {14,4,14}*1568, {14,14,4}*1568a, {14,14,4}*1568c, {14,28,2}*1568c
15-fold covers : {14,6,10}*1680, {14,10,6}*1680, {14,2,30}*1680, {14,30,2}*1680, {42,2,10}*1680, {42,10,2}*1680, {70,2,6}*1680, {70,6,2}*1680, {210,2,2}*1680
16-fold covers : {14,4,8}*1792a, {14,8,4}*1792a, {28,8,2}*1792a, {56,4,2}*1792a, {14,8,8}*1792a, {14,8,8}*1792b, {14,8,8}*1792c, {56,8,2}*1792a, {56,8,2}*1792b, {56,8,2}*1792c, {14,8,8}*1792d, {56,8,2}*1792d, {56,2,8}*1792, {28,4,8}*1792a, {56,4,4}*1792a, {28,4,8}*1792b, {56,4,4}*1792b, {28,8,4}*1792a, {28,4,4}*1792a, {28,4,4}*1792b, {28,8,4}*1792b, {28,8,4}*1792c, {28,8,4}*1792d, {14,4,16}*1792a, {14,16,4}*1792a, {28,16,2}*1792a, {112,4,2}*1792a, {14,4,16}*1792b, {14,16,4}*1792b, {28,16,2}*1792b, {112,4,2}*1792b, {14,4,4}*1792, {14,4,8}*1792b, {14,8,4}*1792b, {28,4,2}*1792, {56,4,2}*1792b, {28,8,2}*1792b, {28,2,16}*1792, {112,2,4}*1792, {14,2,32}*1792, {14,32,2}*1792, {224,2,2}*1792
17-fold covers : {14,2,34}*1904, {14,34,2}*1904, {238,2,2}*1904
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,14);;
s2 := (15,16);;
s3 := (17,18);;
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,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, 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(18)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14);
s1 := Sym(18)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,14);
s2 := Sym(18)!(15,16);
s3 := Sym(18)!(17,18);
poly := sub<Sym(18)|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, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3, 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 >;

```

to this polytope