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# Polytope of Type {7,2}

Atlas Canonical Name : {7,2}*28
if this polytope has a name.
Group : SmallGroup(28,3)
Rank : 3
Schlafli Type : {7,2}
Number of vertices, edges, etc : 7, 7, 2
Order of s0s1s2 : 14
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{7,2,2} of size 56
{7,2,3} of size 84
{7,2,4} of size 112
{7,2,5} of size 140
{7,2,6} of size 168
{7,2,7} of size 196
{7,2,8} of size 224
{7,2,9} of size 252
{7,2,10} of size 280
{7,2,11} of size 308
{7,2,12} of size 336
{7,2,13} of size 364
{7,2,14} of size 392
{7,2,15} of size 420
{7,2,16} of size 448
{7,2,17} of size 476
{7,2,18} of size 504
{7,2,19} of size 532
{7,2,20} of size 560
{7,2,21} of size 588
{7,2,22} of size 616
{7,2,23} of size 644
{7,2,24} of size 672
{7,2,25} of size 700
{7,2,26} of size 728
{7,2,27} of size 756
{7,2,28} of size 784
{7,2,29} of size 812
{7,2,30} of size 840
{7,2,31} of size 868
{7,2,32} of size 896
{7,2,33} of size 924
{7,2,34} of size 952
{7,2,35} of size 980
{7,2,36} of size 1008
{7,2,37} of size 1036
{7,2,38} of size 1064
{7,2,39} of size 1092
{7,2,40} of size 1120
{7,2,41} of size 1148
{7,2,42} of size 1176
{7,2,43} of size 1204
{7,2,44} of size 1232
{7,2,45} of size 1260
{7,2,46} of size 1288
{7,2,47} of size 1316
{7,2,48} of size 1344
{7,2,49} of size 1372
{7,2,50} of size 1400
{7,2,51} of size 1428
{7,2,52} of size 1456
{7,2,53} of size 1484
{7,2,54} of size 1512
{7,2,55} of size 1540
{7,2,56} of size 1568
{7,2,57} of size 1596
{7,2,58} of size 1624
{7,2,59} of size 1652
{7,2,60} of size 1680
{7,2,61} of size 1708
{7,2,62} of size 1736
{7,2,63} of size 1764
{7,2,64} of size 1792
{7,2,65} of size 1820
{7,2,66} of size 1848
{7,2,67} of size 1876
{7,2,68} of size 1904
{7,2,69} of size 1932
{7,2,70} of size 1960
{7,2,71} of size 1988
Vertex Figure Of :
{2,7,2} of size 56
{14,7,2} of size 392
{3,7,2} of size 672
{4,7,2} of size 672
{6,7,2} of size 672
{7,7,2} of size 672
{8,7,2} of size 672
{8,7,2} of size 672
{3,7,2} of size 1008
{7,7,2} of size 1008
{7,7,2} of size 1008
{9,7,2} of size 1008
{9,7,2} of size 1008
{9,7,2} of size 1008
{4,7,2} of size 1344
{6,7,2} of size 1344
{6,7,2} of size 1344
{8,7,2} of size 1344
{8,7,2} of size 1344
{14,7,2} of size 1344
{4,7,2} of size 1792
{7,7,2} of size 1792
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {14,2}*56
3-fold covers : {21,2}*84
4-fold covers : {28,2}*112, {14,4}*112
5-fold covers : {35,2}*140
6-fold covers : {14,6}*168, {42,2}*168
7-fold covers : {49,2}*196, {7,14}*196
8-fold covers : {28,4}*224, {56,2}*224, {14,8}*224
9-fold covers : {63,2}*252, {21,6}*252
10-fold covers : {14,10}*280, {70,2}*280
11-fold covers : {77,2}*308
12-fold covers : {14,12}*336, {28,6}*336a, {84,2}*336, {42,4}*336a, {21,6}*336, {21,4}*336
13-fold covers : {91,2}*364
14-fold covers : {98,2}*392, {14,14}*392a, {14,14}*392c
15-fold covers : {105,2}*420
16-fold covers : {56,4}*448a, {28,4}*448, {56,4}*448b, {28,8}*448a, {28,8}*448b, {112,2}*448, {14,16}*448
17-fold covers : {119,2}*476
18-fold covers : {14,18}*504, {126,2}*504, {42,6}*504a, {42,6}*504b, {42,6}*504c
19-fold covers : {133,2}*532
20-fold covers : {14,20}*560, {28,10}*560, {140,2}*560, {70,4}*560
21-fold covers : {147,2}*588, {21,14}*588
22-fold covers : {14,22}*616, {154,2}*616
23-fold covers : {161,2}*644
24-fold covers : {14,24}*672, {56,6}*672, {28,12}*672, {84,4}*672a, {168,2}*672, {42,8}*672, {21,12}*672, {21,8}*672, {28,6}*672, {42,6}*672, {42,4}*672
25-fold covers : {175,2}*700, {35,10}*700
26-fold covers : {14,26}*728, {182,2}*728
27-fold covers : {189,2}*756, {63,6}*756, {21,6}*756
28-fold covers : {196,2}*784, {98,4}*784, {14,28}*784a, {28,14}*784a, {28,14}*784b, {14,28}*784c
29-fold covers : {203,2}*812
30-fold covers : {14,30}*840, {42,10}*840, {70,6}*840, {210,2}*840
31-fold covers : {217,2}*868
32-fold covers : {56,4}*896a, {56,8}*896a, {56,8}*896b, {28,8}*896a, {56,8}*896c, {56,8}*896d, {112,4}*896a, {112,4}*896b, {28,4}*896, {56,4}*896b, {28,8}*896b, {28,16}*896a, {28,16}*896b, {224,2}*896, {14,32}*896
33-fold covers : {231,2}*924
34-fold covers : {14,34}*952, {238,2}*952
35-fold covers : {245,2}*980, {35,14}*980
36-fold covers : {14,36}*1008, {28,18}*1008a, {252,2}*1008, {126,4}*1008a, {63,4}*1008, {84,6}*1008a, {42,12}*1008a, {42,12}*1008b, {84,6}*1008b, {84,6}*1008c, {42,12}*1008c, {28,4}*1008, {42,4}*1008, {21,12}*1008, {21,6}*1008b, {28,6}*1008
37-fold covers : {259,2}*1036
38-fold covers : {14,38}*1064, {266,2}*1064
39-fold covers : {273,2}*1092
40-fold covers : {14,40}*1120, {56,10}*1120, {28,20}*1120, {140,4}*1120, {280,2}*1120, {70,8}*1120
41-fold covers : {287,2}*1148
42-fold covers : {98,6}*1176, {294,2}*1176, {14,42}*1176a, {14,42}*1176b, {42,14}*1176b, {42,14}*1176c
43-fold covers : {301,2}*1204
44-fold covers : {28,22}*1232, {14,44}*1232, {308,2}*1232, {154,4}*1232
45-fold covers : {315,2}*1260, {105,6}*1260
46-fold covers : {14,46}*1288, {322,2}*1288
47-fold covers : {329,2}*1316
48-fold covers : {14,48}*1344, {112,6}*1344, {28,12}*1344a, {28,24}*1344a, {56,12}*1344a, {28,24}*1344b, {56,12}*1344b, {168,4}*1344a, {84,4}*1344a, {168,4}*1344b, {84,8}*1344a, {84,8}*1344b, {336,2}*1344, {42,16}*1344, {21,6}*1344, {21,8}*1344, {28,12}*1344b, {28,6}*1344e, {84,6}*1344a, {42,12}*1344a, {42,6}*1344, {56,6}*1344b, {56,6}*1344c, {84,6}*1344b, {28,12}*1344c, {42,12}*1344b, {84,4}*1344b, {42,4}*1344b, {84,4}*1344c, {42,8}*1344b, {42,8}*1344c
49-fold covers : {343,2}*1372, {49,14}*1372, {7,14}*1372
50-fold covers : {14,50}*1400, {350,2}*1400, {70,10}*1400a, {70,10}*1400b, {70,10}*1400c
51-fold covers : {357,2}*1428
52-fold covers : {28,26}*1456, {14,52}*1456, {364,2}*1456, {182,4}*1456
53-fold covers : {371,2}*1484
54-fold covers : {14,54}*1512, {378,2}*1512, {42,18}*1512a, {42,6}*1512a, {126,6}*1512a, {126,6}*1512b, {42,18}*1512b, {42,6}*1512b, {42,6}*1512c, {42,6}*1512d
55-fold covers : {385,2}*1540
56-fold covers : {196,4}*1568, {392,2}*1568, {98,8}*1568, {14,56}*1568a, {56,14}*1568a, {56,14}*1568b, {28,28}*1568a, {28,28}*1568c, {14,56}*1568c
57-fold covers : {399,2}*1596
58-fold covers : {14,58}*1624, {406,2}*1624
59-fold covers : {413,2}*1652
60-fold covers : {14,60}*1680, {28,30}*1680a, {42,20}*1680a, {84,10}*1680, {70,12}*1680, {140,6}*1680a, {420,2}*1680, {210,4}*1680a, {35,4}*1680, {35,6}*1680b, {21,10}*1680, {35,6}*1680c, {35,10}*1680, {105,6}*1680, {105,4}*1680
61-fold covers : {427,2}*1708
62-fold covers : {14,62}*1736, {434,2}*1736
63-fold covers : {441,2}*1764, {147,6}*1764, {63,14}*1764, {21,42}*1764
64-fold covers : {56,8}*1792a, {28,8}*1792a, {56,8}*1792b, {56,4}*1792a, {56,8}*1792c, {56,8}*1792d, {28,16}*1792a, {112,4}*1792a, {28,16}*1792b, {112,4}*1792b, {112,8}*1792a, {56,16}*1792a, {112,8}*1792b, {56,16}*1792b, {56,16}*1792c, {112,8}*1792c, {112,8}*1792d, {56,16}*1792d, {56,16}*1792e, {112,8}*1792e, {112,8}*1792f, {56,16}*1792f, {28,32}*1792a, {224,4}*1792a, {28,32}*1792b, {224,4}*1792b, {28,4}*1792, {56,4}*1792b, {28,8}*1792b, {28,8}*1792c, {56,8}*1792e, {56,4}*1792c, {56,4}*1792d, {28,8}*1792d, {56,8}*1792f, {56,8}*1792g, {56,8}*1792h, {14,64}*1792, {448,2}*1792, {7,4}*1792c
65-fold covers : {455,2}*1820
66-fold covers : {42,22}*1848, {14,66}*1848, {154,6}*1848, {462,2}*1848
67-fold covers : {469,2}*1876
68-fold covers : {28,34}*1904, {14,68}*1904, {476,2}*1904, {238,4}*1904
69-fold covers : {483,2}*1932
70-fold covers : {98,10}*1960, {490,2}*1960, {14,70}*1960a, {14,70}*1960b, {70,14}*1960b, {70,14}*1960c
71-fold covers : {497,2}*1988
Permutation Representation (GAP) :
```s0 := (2,3)(4,5)(6,7);;
s1 := (1,2)(3,4)(5,6);;
s2 := (8,9);;
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, s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(9)!(2,3)(4,5)(6,7);
s1 := Sym(9)!(1,2)(3,4)(5,6);
s2 := Sym(9)!(8,9);
poly := sub<Sym(9)|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, s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope