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# Polytope of Type {4,14,4}

Atlas Canonical Name : {4,14,4}*448
Also Known As : {{4,14|2},{14,4|2}}. if this polytope has another name.
Group : SmallGroup(448,1043)
Rank : 4
Schlafli Type : {4,14,4}
Number of vertices, edges, etc : 4, 28, 28, 4
Order of s0s1s2s3 : 28
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,14,4,2} of size 896
{4,14,4,4} of size 1792
Vertex Figure Of :
{2,4,14,4} of size 896
{4,4,14,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,14,4}*224, {4,14,2}*224
4-fold quotients : {2,14,2}*112
7-fold quotients : {4,2,4}*64
8-fold quotients : {2,7,2}*56
14-fold quotients : {2,2,4}*32, {4,2,2}*32
28-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,28,4}*896, {4,14,8}*896, {8,14,4}*896
3-fold covers : {4,14,12}*1344, {12,14,4}*1344, {4,42,4}*1344a
4-fold covers : {8,14,8}*1792, {4,28,8}*1792a, {8,28,4}*1792a, {4,28,8}*1792b, {8,28,4}*1792b, {4,56,4}*1792a, {4,28,4}*1792a, {4,28,4}*1792b, {4,56,4}*1792b, {4,56,4}*1792c, {4,56,4}*1792d, {4,14,16}*1792, {16,14,4}*1792
Permutation Representation (GAP) :
```s0 := ( 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, 99)( 86,100)
( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)( 94,108)
( 95,109)( 96,110)( 97,111)( 98,112);;
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, 99)( 30,105)( 31,104)( 32,103)
( 33,102)( 34,101)( 35,100)( 36,106)( 37,112)( 38,111)( 39,110)( 40,109)
( 41,108)( 42,107)( 43, 85)( 44, 91)( 45, 90)( 46, 89)( 47, 88)( 48, 87)
( 49, 86)( 50, 92)( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)( 56, 93);;
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);;
s3 := (  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, 43)( 16, 44)
( 17, 45)( 18, 46)( 19, 47)( 20, 48)( 21, 49)( 22, 50)( 23, 51)( 24, 52)
( 25, 53)( 26, 54)( 27, 55)( 28, 56)( 57, 99)( 58,100)( 59,101)( 60,102)
( 61,103)( 62,104)( 63,105)( 64,106)( 65,107)( 66,108)( 67,109)( 68,110)
( 69,111)( 70,112)( 71, 85)( 72, 86)( 73, 87)( 74, 88)( 75, 89)( 76, 90)
( 77, 91)( 78, 92)( 79, 93)( 80, 94)( 81, 95)( 82, 96)( 83, 97)( 84, 98);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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)!( 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, 99)
( 86,100)( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)
( 94,108)( 95,109)( 96,110)( 97,111)( 98,112);
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, 99)( 30,105)( 31,104)
( 32,103)( 33,102)( 34,101)( 35,100)( 36,106)( 37,112)( 38,111)( 39,110)
( 40,109)( 41,108)( 42,107)( 43, 85)( 44, 91)( 45, 90)( 46, 89)( 47, 88)
( 48, 87)( 49, 86)( 50, 92)( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)
( 56, 93);
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);
s3 := Sym(112)!(  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, 43)
( 16, 44)( 17, 45)( 18, 46)( 19, 47)( 20, 48)( 21, 49)( 22, 50)( 23, 51)
( 24, 52)( 25, 53)( 26, 54)( 27, 55)( 28, 56)( 57, 99)( 58,100)( 59,101)
( 60,102)( 61,103)( 62,104)( 63,105)( 64,106)( 65,107)( 66,108)( 67,109)
( 68,110)( 69,111)( 70,112)( 71, 85)( 72, 86)( 73, 87)( 74, 88)( 75, 89)
( 76, 90)( 77, 91)( 78, 92)( 79, 93)( 80, 94)( 81, 95)( 82, 96)( 83, 97)
( 84, 98);
poly := sub<Sym(112)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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