# Polytope of Type {8,28}

Atlas Canonical Name : {8,28}*896b
if this polytope has a name.
Group : SmallGroup(896,1666)
Rank : 3
Schlafli Type : {8,28}
Number of vertices, edges, etc : 16, 224, 56
Order of s0s1s2 : 28
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,28,2} of size 1792
Vertex Figure Of :
{2,8,28} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,28}*448
4-fold quotients : {4,28}*224
7-fold quotients : {8,4}*128b
8-fold quotients : {2,28}*112, {4,14}*112
14-fold quotients : {4,4}*64
16-fold quotients : {2,14}*56
28-fold quotients : {4,4}*32
32-fold quotients : {2,7}*28
56-fold quotients : {2,4}*16, {4,2}*16
112-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,28}*1792a, {8,56}*1792b, {8,56}*1792d, {8,28}*1792b, {8,28}*1792d, {8,56}*1792f, {8,56}*1792h
Permutation Representation (GAP) :
```s0 := (  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)(  8, 64)
(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)
( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)( 24, 80)
( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29,106)( 30,107)( 31,108)( 32,109)
( 33,110)( 34,111)( 35,112)( 36, 99)( 37,100)( 38,101)( 39,102)( 40,103)
( 41,104)( 42,105)( 43, 92)( 44, 93)( 45, 94)( 46, 95)( 47, 96)( 48, 97)
( 49, 98)( 50, 85)( 51, 86)( 52, 87)( 53, 88)( 54, 89)( 55, 90)( 56, 91);;
s1 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 15, 22)( 16, 28)
( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)( 46, 54)( 47, 53)
( 48, 52)( 49, 51)( 57, 85)( 58, 91)( 59, 90)( 60, 89)( 61, 88)( 62, 87)
( 63, 86)( 64, 92)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)( 70, 93)
( 71,106)( 72,112)( 73,111)( 74,110)( 75,109)( 76,108)( 77,107)( 78, 99)
( 79,105)( 80,104)( 81,103)( 82,102)( 83,101)( 84,100);;
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, 51)( 30, 50)( 31, 56)( 32, 55)
( 33, 54)( 34, 53)( 35, 52)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)
( 41, 46)( 42, 45)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)
( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,107)( 86,106)
( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,100)( 93, 99)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101);;
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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*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(112)!(  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)
(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)
( 16, 72)( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)
( 24, 80)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29,106)( 30,107)( 31,108)
( 32,109)( 33,110)( 34,111)( 35,112)( 36, 99)( 37,100)( 38,101)( 39,102)
( 40,103)( 41,104)( 42,105)( 43, 92)( 44, 93)( 45, 94)( 46, 95)( 47, 96)
( 48, 97)( 49, 98)( 50, 85)( 51, 86)( 52, 87)( 53, 88)( 54, 89)( 55, 90)
( 56, 91);
s1 := Sym(112)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 15, 22)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 30, 35)( 31, 34)
( 32, 33)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 57, 85)( 58, 91)( 59, 90)( 60, 89)( 61, 88)
( 62, 87)( 63, 86)( 64, 92)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)
( 70, 93)( 71,106)( 72,112)( 73,111)( 74,110)( 75,109)( 76,108)( 77,107)
( 78, 99)( 79,105)( 80,104)( 81,103)( 82,102)( 83,101)( 84,100);
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, 51)( 30, 50)( 31, 56)
( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 44)( 37, 43)( 38, 49)( 39, 48)
( 40, 47)( 41, 46)( 42, 45)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)
( 67, 69)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,107)
( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,100)( 93, 99)
( 94,105)( 95,104)( 96,103)( 97,102)( 98,101);
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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*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