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# Polytope of Type {40,4}

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

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

```
References : None.
to this polytope