# Polytope of Type {4,40}

Atlas Canonical Name : {4,40}*1280h
if this polytope has a name.
Group : SmallGroup(1280,1116434)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 16, 320, 160
Order of s0s1s2 : 40
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,20}*640c
4-fold quotients : {4,10}*320a
8-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,108)(  2,107)(  3,105)(  4,106)(  5,112)(  6,111)(  7,109)(  8,110)
(  9, 99)( 10,100)( 11, 98)( 12, 97)( 13,103)( 14,104)( 15,102)( 16,101)
( 17,124)( 18,123)( 19,121)( 20,122)( 21,128)( 22,127)( 23,125)( 24,126)
( 25,115)( 26,116)( 27,114)( 28,113)( 29,119)( 30,120)( 31,118)( 32,117)
( 33, 76)( 34, 75)( 35, 73)( 36, 74)( 37, 80)( 38, 79)( 39, 77)( 40, 78)
( 41, 67)( 42, 68)( 43, 66)( 44, 65)( 45, 71)( 46, 72)( 47, 70)( 48, 69)
( 49, 92)( 50, 91)( 51, 89)( 52, 90)( 53, 96)( 54, 95)( 55, 93)( 56, 94)
( 57, 83)( 58, 84)( 59, 82)( 60, 81)( 61, 87)( 62, 88)( 63, 86)( 64, 85);;
s1 := (  3,  4)(  5,  7)(  6,  8)(  9, 65)( 10, 66)( 11, 68)( 12, 67)( 13, 71)
( 14, 72)( 15, 69)( 16, 70)( 17, 90)( 18, 89)( 19, 91)( 20, 92)( 21, 96)
( 22, 95)( 23, 94)( 24, 93)( 25, 26)( 29, 32)( 30, 31)( 33,114)( 34,113)
( 35,115)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)( 41, 50)( 42, 49)
( 43, 51)( 44, 52)( 45, 56)( 46, 55)( 47, 54)( 48, 53)( 57,106)( 58,105)
( 59,107)( 60,108)( 61,112)( 62,111)( 63,110)( 64,109)( 73, 74)( 77, 80)
( 78, 79)( 83, 84)( 85, 87)( 86, 88)( 97,121)( 98,122)( 99,124)(100,123)
(101,127)(102,128)(103,125)(104,126);;
s2 := (  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9,103)( 10,104)( 11,101)( 12,102)
( 13, 99)( 14,100)( 15, 97)( 16, 98)( 17, 63)( 18, 64)( 19, 61)( 20, 62)
( 21, 59)( 22, 60)( 23, 57)( 24, 58)( 25, 96)( 26, 95)( 27, 94)( 28, 93)
( 29, 92)( 30, 91)( 31, 90)( 32, 89)( 33, 79)( 34, 80)( 35, 77)( 36, 78)
( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 48)( 42, 47)( 43, 46)( 44, 45)
( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)( 56,114)
( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 81,127)( 82,128)( 83,125)( 84,126)
( 85,123)( 86,124)( 87,121)( 88,122)(105,112)(106,111)(107,110)(108,109);;
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,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(128)!(  1,108)(  2,107)(  3,105)(  4,106)(  5,112)(  6,111)(  7,109)
(  8,110)(  9, 99)( 10,100)( 11, 98)( 12, 97)( 13,103)( 14,104)( 15,102)
( 16,101)( 17,124)( 18,123)( 19,121)( 20,122)( 21,128)( 22,127)( 23,125)
( 24,126)( 25,115)( 26,116)( 27,114)( 28,113)( 29,119)( 30,120)( 31,118)
( 32,117)( 33, 76)( 34, 75)( 35, 73)( 36, 74)( 37, 80)( 38, 79)( 39, 77)
( 40, 78)( 41, 67)( 42, 68)( 43, 66)( 44, 65)( 45, 71)( 46, 72)( 47, 70)
( 48, 69)( 49, 92)( 50, 91)( 51, 89)( 52, 90)( 53, 96)( 54, 95)( 55, 93)
( 56, 94)( 57, 83)( 58, 84)( 59, 82)( 60, 81)( 61, 87)( 62, 88)( 63, 86)
( 64, 85);
s1 := Sym(128)!(  3,  4)(  5,  7)(  6,  8)(  9, 65)( 10, 66)( 11, 68)( 12, 67)
( 13, 71)( 14, 72)( 15, 69)( 16, 70)( 17, 90)( 18, 89)( 19, 91)( 20, 92)
( 21, 96)( 22, 95)( 23, 94)( 24, 93)( 25, 26)( 29, 32)( 30, 31)( 33,114)
( 34,113)( 35,115)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)( 41, 50)
( 42, 49)( 43, 51)( 44, 52)( 45, 56)( 46, 55)( 47, 54)( 48, 53)( 57,106)
( 58,105)( 59,107)( 60,108)( 61,112)( 62,111)( 63,110)( 64,109)( 73, 74)
( 77, 80)( 78, 79)( 83, 84)( 85, 87)( 86, 88)( 97,121)( 98,122)( 99,124)
(100,123)(101,127)(102,128)(103,125)(104,126);
s2 := Sym(128)!(  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9,103)( 10,104)( 11,101)
( 12,102)( 13, 99)( 14,100)( 15, 97)( 16, 98)( 17, 63)( 18, 64)( 19, 61)
( 20, 62)( 21, 59)( 22, 60)( 23, 57)( 24, 58)( 25, 96)( 26, 95)( 27, 94)
( 28, 93)( 29, 92)( 30, 91)( 31, 90)( 32, 89)( 33, 79)( 34, 80)( 35, 77)
( 36, 78)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 48)( 42, 47)( 43, 46)
( 44, 45)( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)
( 56,114)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 81,127)( 82,128)( 83,125)
( 84,126)( 85,123)( 86,124)( 87,121)( 88,122)(105,112)(106,111)(107,110)
(108,109);
poly := sub<Sym(128)|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,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope