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# Polytope of Type {4,14}

Atlas Canonical Name : {4,14}*112
Also Known As : {4,14|2}. if this polytope has another name.
Group : SmallGroup(112,31)
Rank : 3
Schlafli Type : {4,14}
Number of vertices, edges, etc : 4, 28, 14
Order of s0s1s2 : 28
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,14,2} of size 224
{4,14,4} of size 448
{4,14,6} of size 672
{4,14,7} of size 784
{4,14,8} of size 896
{4,14,10} of size 1120
{4,14,12} of size 1344
{4,14,14} of size 1568
{4,14,14} of size 1568
{4,14,14} of size 1568
{4,14,16} of size 1792
Vertex Figure Of :
{2,4,14} of size 224
{4,4,14} of size 448
{6,4,14} of size 672
{3,4,14} of size 672
{8,4,14} of size 896
{8,4,14} of size 896
{4,4,14} of size 896
{6,4,14} of size 1008
{10,4,14} of size 1120
{12,4,14} of size 1344
{6,4,14} of size 1344
{14,4,14} of size 1568
{5,4,14} of size 1680
{8,4,14} of size 1792
{16,4,14} of size 1792
{16,4,14} of size 1792
{4,4,14} of size 1792
{8,4,14} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,14}*56
4-fold quotients : {2,7}*28
7-fold quotients : {4,2}*16
14-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,28}*224, {8,14}*224
3-fold covers : {12,14}*336, {4,42}*336a
4-fold covers : {4,56}*448a, {4,28}*448, {4,56}*448b, {8,28}*448a, {8,28}*448b, {16,14}*448
5-fold covers : {20,14}*560, {4,70}*560
6-fold covers : {24,14}*672, {12,28}*672, {4,84}*672a, {8,42}*672
7-fold covers : {4,98}*784, {28,14}*784a, {28,14}*784c
8-fold covers : {4,56}*896a, {8,56}*896a, {8,56}*896b, {8,28}*896a, {8,56}*896c, {8,56}*896d, {4,112}*896a, {4,112}*896b, {4,28}*896, {4,56}*896b, {8,28}*896b, {16,28}*896a, {16,28}*896b, {32,14}*896
9-fold covers : {36,14}*1008, {4,126}*1008a, {12,42}*1008a, {12,42}*1008b, {12,42}*1008c, {4,42}*1008
10-fold covers : {40,14}*1120, {20,28}*1120, {4,140}*1120, {8,70}*1120
11-fold covers : {44,14}*1232, {4,154}*1232
12-fold covers : {48,14}*1344, {12,28}*1344a, {24,28}*1344a, {12,56}*1344a, {24,28}*1344b, {12,56}*1344b, {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b, {16,42}*1344, {12,28}*1344b, {12,42}*1344b, {4,42}*1344b
13-fold covers : {52,14}*1456, {4,182}*1456
14-fold covers : {4,196}*1568, {8,98}*1568, {56,14}*1568a, {28,28}*1568a, {28,28}*1568b, {56,14}*1568c
15-fold covers : {60,14}*1680, {20,42}*1680a, {12,70}*1680, {4,210}*1680a
16-fold covers : {8,56}*1792a, {8,28}*1792a, {8,56}*1792b, {4,56}*1792a, {8,56}*1792c, {8,56}*1792d, {16,28}*1792a, {4,112}*1792a, {16,28}*1792b, {4,112}*1792b, {8,112}*1792a, {16,56}*1792a, {8,112}*1792b, {16,56}*1792b, {16,56}*1792c, {8,112}*1792c, {8,112}*1792d, {16,56}*1792d, {16,56}*1792e, {8,112}*1792e, {8,112}*1792f, {16,56}*1792f, {32,28}*1792a, {4,224}*1792a, {32,28}*1792b, {4,224}*1792b, {4,28}*1792, {4,56}*1792b, {8,28}*1792b, {8,28}*1792c, {8,56}*1792e, {4,56}*1792c, {4,56}*1792d, {8,28}*1792d, {8,56}*1792f, {8,56}*1792g, {8,56}*1792h, {64,14}*1792
17-fold covers : {68,14}*1904, {4,238}*1904
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 6,11)( 7,12)(13,19)(14,20)(21,25)(22,26);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 5,10)( 8,14)( 9,13)(11,18)(12,17)(15,22)(16,21)
(19,24)(20,23)(25,28)(26,27);;
s2 := ( 1, 3)( 2, 6)( 4, 8)( 5,11)( 7,13)( 9,15)(10,17)(12,19)(14,21)(18,23)
(20,25)(24,27);;
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*s0*s1*s0*s1*s0*s1,
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(28)!( 2, 5)( 6,11)( 7,12)(13,19)(14,20)(21,25)(22,26);
s1 := Sym(28)!( 1, 2)( 3, 7)( 4, 6)( 5,10)( 8,14)( 9,13)(11,18)(12,17)(15,22)
(16,21)(19,24)(20,23)(25,28)(26,27);
s2 := Sym(28)!( 1, 3)( 2, 6)( 4, 8)( 5,11)( 7,13)( 9,15)(10,17)(12,19)(14,21)
(18,23)(20,25)(24,27);
poly := sub<Sym(28)|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*s0*s1*s0*s1*s0*s1,
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 >;

```
References : None.
to this polytope