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# Polytope of Type {14,8}

Atlas Canonical Name : {14,8}*224
Also Known As : {14,8|2}. if this polytope has another name.
Group : SmallGroup(224,105)
Rank : 3
Schlafli Type : {14,8}
Number of vertices, edges, etc : 14, 56, 8
Order of s0s1s2 : 56
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{14,8,2} of size 448
{14,8,4} of size 896
{14,8,4} of size 896
{14,8,6} of size 1344
{14,8,3} of size 1344
{14,8,4} of size 1792
{14,8,8} of size 1792
{14,8,8} of size 1792
{14,8,8} of size 1792
{14,8,8} of size 1792
{14,8,4} of size 1792
Vertex Figure Of :
{2,14,8} of size 448
{4,14,8} of size 896
{6,14,8} of size 1344
{7,14,8} of size 1568
{8,14,8} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {14,4}*112
4-fold quotients : {14,2}*56
7-fold quotients : {2,8}*32
8-fold quotients : {7,2}*28
14-fold quotients : {2,4}*16
28-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {28,8}*448a, {14,16}*448
3-fold covers : {14,24}*672, {42,8}*672
4-fold covers : {56,8}*896b, {28,8}*896a, {56,8}*896d, {28,16}*896a, {28,16}*896b, {14,32}*896
5-fold covers : {14,40}*1120, {70,8}*1120
6-fold covers : {14,48}*1344, {28,24}*1344a, {84,8}*1344a, {42,16}*1344
7-fold covers : {98,8}*1568, {14,56}*1568a, {14,56}*1568c
8-fold covers : {56,8}*1792a, {28,8}*1792a, {56,8}*1792c, {28,16}*1792a, {28,16}*1792b, {112,8}*1792a, {112,8}*1792b, {56,16}*1792c, {112,8}*1792d, {56,16}*1792d, {56,16}*1792e, {112,8}*1792f, {56,16}*1792f, {28,32}*1792a, {28,32}*1792b, {14,64}*1792
Permutation Representation (GAP) :
```s0 := ( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)
(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)(46,47)
(51,56)(52,55)(53,54);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,23)(16,22)(17,28)(18,27)
(19,26)(20,25)(21,24)(29,44)(30,43)(31,49)(32,48)(33,47)(34,46)(35,45)(36,51)
(37,50)(38,56)(39,55)(40,54)(41,53)(42,52);;
s2 := ( 1,29)( 2,30)( 3,31)( 4,32)( 5,33)( 6,34)( 7,35)( 8,36)( 9,37)(10,38)
(11,39)(12,40)(13,41)(14,42)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)
(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49);;
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,
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(56)!( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)
(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)
(46,47)(51,56)(52,55)(53,54);
s1 := Sym(56)!( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,23)(16,22)(17,28)
(18,27)(19,26)(20,25)(21,24)(29,44)(30,43)(31,49)(32,48)(33,47)(34,46)(35,45)
(36,51)(37,50)(38,56)(39,55)(40,54)(41,53)(42,52);
s2 := Sym(56)!( 1,29)( 2,30)( 3,31)( 4,32)( 5,33)( 6,34)( 7,35)( 8,36)( 9,37)
(10,38)(11,39)(12,40)(13,41)(14,42)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)
(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49);
poly := sub<Sym(56)|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,
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