# Polytope of Type {28,10}

Atlas Canonical Name : {28,10}*560
Also Known As : {28,10|2}. if this polytope has another name.
Group : SmallGroup(560,121)
Rank : 3
Schlafli Type : {28,10}
Number of vertices, edges, etc : 28, 140, 10
Order of s0s1s2 : 140
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{28,10,2} of size 1120
Vertex Figure Of :
{2,28,10} of size 1120
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {14,10}*280
5-fold quotients : {28,2}*112
7-fold quotients : {4,10}*80
10-fold quotients : {14,2}*56
14-fold quotients : {2,10}*40
20-fold quotients : {7,2}*28
28-fold quotients : {2,5}*20
35-fold quotients : {4,2}*16
70-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {56,10}*1120, {28,20}*1120
3-fold covers : {28,30}*1680a, {84,10}*1680
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)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 71,106)( 72,112)
( 73,111)( 74,110)( 75,109)( 76,108)( 77,107)( 78,113)( 79,119)( 80,118)
( 81,117)( 82,116)( 83,115)( 84,114)( 85,120)( 86,126)( 87,125)( 88,124)
( 89,123)( 90,122)( 91,121)( 92,127)( 93,133)( 94,132)( 95,131)( 96,130)
( 97,129)( 98,128)( 99,134)(100,140)(101,139)(102,138)(103,137)(104,136)
(105,135);;
s1 := (  1, 72)(  2, 71)(  3, 77)(  4, 76)(  5, 75)(  6, 74)(  7, 73)(  8,100)
(  9, 99)( 10,105)( 11,104)( 12,103)( 13,102)( 14,101)( 15, 93)( 16, 92)
( 17, 98)( 18, 97)( 19, 96)( 20, 95)( 21, 94)( 22, 86)( 23, 85)( 24, 91)
( 25, 90)( 26, 89)( 27, 88)( 28, 87)( 29, 79)( 30, 78)( 31, 84)( 32, 83)
( 33, 82)( 34, 81)( 35, 80)( 36,107)( 37,106)( 38,112)( 39,111)( 40,110)
( 41,109)( 42,108)( 43,135)( 44,134)( 45,140)( 46,139)( 47,138)( 48,137)
( 49,136)( 50,128)( 51,127)( 52,133)( 53,132)( 54,131)( 55,130)( 56,129)
( 57,121)( 58,120)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)( 64,114)
( 65,113)( 66,119)( 67,118)( 68,117)( 69,116)( 70,115);;
s2 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 15, 29)
( 16, 30)( 17, 31)( 18, 32)( 19, 33)( 20, 34)( 21, 35)( 36, 43)( 37, 44)
( 38, 45)( 39, 46)( 40, 47)( 41, 48)( 42, 49)( 50, 64)( 51, 65)( 52, 66)
( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 71, 78)( 72, 79)( 73, 80)( 74, 81)
( 75, 82)( 76, 83)( 77, 84)( 85, 99)( 86,100)( 87,101)( 88,102)( 89,103)
( 90,104)( 91,105)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)
(112,119)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140);;
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*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*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(140)!(  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)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 71,106)
( 72,112)( 73,111)( 74,110)( 75,109)( 76,108)( 77,107)( 78,113)( 79,119)
( 80,118)( 81,117)( 82,116)( 83,115)( 84,114)( 85,120)( 86,126)( 87,125)
( 88,124)( 89,123)( 90,122)( 91,121)( 92,127)( 93,133)( 94,132)( 95,131)
( 96,130)( 97,129)( 98,128)( 99,134)(100,140)(101,139)(102,138)(103,137)
(104,136)(105,135);
s1 := Sym(140)!(  1, 72)(  2, 71)(  3, 77)(  4, 76)(  5, 75)(  6, 74)(  7, 73)
(  8,100)(  9, 99)( 10,105)( 11,104)( 12,103)( 13,102)( 14,101)( 15, 93)
( 16, 92)( 17, 98)( 18, 97)( 19, 96)( 20, 95)( 21, 94)( 22, 86)( 23, 85)
( 24, 91)( 25, 90)( 26, 89)( 27, 88)( 28, 87)( 29, 79)( 30, 78)( 31, 84)
( 32, 83)( 33, 82)( 34, 81)( 35, 80)( 36,107)( 37,106)( 38,112)( 39,111)
( 40,110)( 41,109)( 42,108)( 43,135)( 44,134)( 45,140)( 46,139)( 47,138)
( 48,137)( 49,136)( 50,128)( 51,127)( 52,133)( 53,132)( 54,131)( 55,130)
( 56,129)( 57,121)( 58,120)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)
( 64,114)( 65,113)( 66,119)( 67,118)( 68,117)( 69,116)( 70,115);
s2 := Sym(140)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 15, 29)( 16, 30)( 17, 31)( 18, 32)( 19, 33)( 20, 34)( 21, 35)( 36, 43)
( 37, 44)( 38, 45)( 39, 46)( 40, 47)( 41, 48)( 42, 49)( 50, 64)( 51, 65)
( 52, 66)( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85, 99)( 86,100)( 87,101)( 88,102)
( 89,103)( 90,104)( 91,105)(106,113)(107,114)(108,115)(109,116)(110,117)
(111,118)(112,119)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)
(126,140);
poly := sub<Sym(140)|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*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*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