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# Polytope of Type {4,56}

Atlas Canonical Name : {4,56}*448b
if this polytope has a name.
Group : SmallGroup(448,281)
Rank : 3
Schlafli Type : {4,56}
Number of vertices, edges, etc : 4, 112, 56
Order of s0s1s2 : 56
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,56,2} of size 896
{4,56,4} of size 1792
{4,56,4} of size 1792
Vertex Figure Of :
{2,4,56} of size 896
{4,4,56} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,28}*224
4-fold quotients : {2,28}*112, {4,14}*112
7-fold quotients : {4,8}*64b
8-fold quotients : {2,14}*56
14-fold quotients : {4,4}*32
16-fold quotients : {2,7}*28
28-fold quotients : {2,4}*16, {4,2}*16
56-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,56}*896a, {8,56}*896c, {8,56}*896d
3-fold covers : {12,56}*1344b, {4,168}*1344b
4-fold covers : {8,56}*1792a, {4,56}*1792a, {8,56}*1792d, {4,112}*1792a, {4,112}*1792b, {8,112}*1792a, {8,112}*1792b, {16,56}*1792c, {16,56}*1792e
Permutation Representation (GAP) :
```s0 := ( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 57, 71)( 58, 72)
( 59, 73)( 60, 74)( 61, 75)( 62, 76)( 63, 77)( 64, 78)( 65, 79)( 66, 80)
( 67, 81)( 68, 82)( 69, 83)( 70, 84)( 85,106)( 86,107)( 87,108)( 88,109)
( 89,110)( 90,111)( 91,112)( 92, 99)( 93,100)( 94,101)( 95,102)( 96,103)
( 97,104)( 98,105);;
s1 := (  1, 57)(  2, 63)(  3, 62)(  4, 61)(  5, 60)(  6, 59)(  7, 58)(  8, 64)
(  9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 71)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 78)( 23, 84)( 24, 83)
( 25, 82)( 26, 81)( 27, 80)( 28, 79)( 29, 92)( 30, 98)( 31, 97)( 32, 96)
( 33, 95)( 34, 94)( 35, 93)( 36, 85)( 37, 91)( 38, 90)( 39, 89)( 40, 88)
( 41, 87)( 42, 86)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)( 48,108)
( 49,107)( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)( 56,100);;
s2 := (  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, 37)( 30, 36)( 31, 42)
( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 44)( 45, 49)( 46, 48)( 50, 51)
( 52, 56)( 53, 55)( 57, 86)( 58, 85)( 59, 91)( 60, 90)( 61, 89)( 62, 88)
( 63, 87)( 64, 93)( 65, 92)( 66, 98)( 67, 97)( 68, 96)( 69, 95)( 70, 94)
( 71,107)( 72,106)( 73,112)( 74,111)( 75,110)( 76,109)( 77,108)( 78,100)
( 79, 99)( 80,105)( 81,104)( 82,103)( 83,102)( 84,101);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(112)!( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)
( 43, 50)( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 57, 71)
( 58, 72)( 59, 73)( 60, 74)( 61, 75)( 62, 76)( 63, 77)( 64, 78)( 65, 79)
( 66, 80)( 67, 81)( 68, 82)( 69, 83)( 70, 84)( 85,106)( 86,107)( 87,108)
( 88,109)( 89,110)( 90,111)( 91,112)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105);
s1 := Sym(112)!(  1, 57)(  2, 63)(  3, 62)(  4, 61)(  5, 60)(  6, 59)(  7, 58)
(  8, 64)(  9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 71)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 78)( 23, 84)
( 24, 83)( 25, 82)( 26, 81)( 27, 80)( 28, 79)( 29, 92)( 30, 98)( 31, 97)
( 32, 96)( 33, 95)( 34, 94)( 35, 93)( 36, 85)( 37, 91)( 38, 90)( 39, 89)
( 40, 88)( 41, 87)( 42, 86)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)
( 48,108)( 49,107)( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)
( 56,100);
s2 := Sym(112)!(  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, 37)( 30, 36)
( 31, 42)( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 44)( 45, 49)( 46, 48)
( 50, 51)( 52, 56)( 53, 55)( 57, 86)( 58, 85)( 59, 91)( 60, 90)( 61, 89)
( 62, 88)( 63, 87)( 64, 93)( 65, 92)( 66, 98)( 67, 97)( 68, 96)( 69, 95)
( 70, 94)( 71,107)( 72,106)( 73,112)( 74,111)( 75,110)( 76,109)( 77,108)
( 78,100)( 79, 99)( 80,105)( 81,104)( 82,103)( 83,102)( 84,101);
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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 >;

```
References : None.
to this polytope