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# Polytope of Type {4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*1280d
if this polytope has a name.
Group : SmallGroup(1280,1116454)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 32, 320, 160
Order of s0s1s2 : 10
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {4,20}*640d, {4,20}*640e, {4,10}*640b
4-fold quotients : {4,5}*320, {4,10}*320a, {4,10}*320b
8-fold quotients : {4,5}*160
32-fold quotients : {2,10}*40
64-fold quotients : {2,5}*20
160-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 60)(  2, 59)(  3, 57)(  4, 58)(  5, 64)(  6, 63)(  7, 61)(  8, 62)
(  9, 52)( 10, 51)( 11, 49)( 12, 50)( 13, 56)( 14, 55)( 15, 53)( 16, 54)
( 17, 44)( 18, 43)( 19, 41)( 20, 42)( 21, 48)( 22, 47)( 23, 45)( 24, 46)
( 25, 36)( 26, 35)( 27, 33)( 28, 34)( 29, 40)( 30, 39)( 31, 37)( 32, 38)
( 65,124)( 66,123)( 67,121)( 68,122)( 69,128)( 70,127)( 71,125)( 72,126)
( 73,116)( 74,115)( 75,113)( 76,114)( 77,120)( 78,119)( 79,117)( 80,118)
( 81,108)( 82,107)( 83,105)( 84,106)( 85,112)( 86,111)( 87,109)( 88,110)
( 89,100)( 90, 99)( 91, 97)( 92, 98)( 93,104)( 94,103)( 95,101)( 96,102);;
s1 := (  3,  4)(  7,  8)(  9, 65)( 10, 66)( 11, 68)( 12, 67)( 13, 69)( 14, 70)
( 15, 72)( 16, 71)( 17, 90)( 18, 89)( 19, 91)( 20, 92)( 21, 94)( 22, 93)
( 23, 95)( 24, 96)( 25, 26)( 29, 30)( 33,113)( 34,114)( 35,116)( 36,115)
( 37,117)( 38,118)( 39,120)( 40,119)( 41, 49)( 42, 50)( 43, 52)( 44, 51)
( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 57,105)( 58,106)( 59,108)( 60,107)
( 61,109)( 62,110)( 63,112)( 64,111)( 73, 74)( 77, 78)( 83, 84)( 87, 88)
( 97,122)( 98,121)( 99,123)(100,124)(101,126)(102,125)(103,127)(104,128);;
s2 := (  1,  8)(  2,  7)(  3,  6)(  4,  5)(  9, 16)( 10, 15)( 11, 14)( 12, 13)
( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 27)( 22, 28)( 23, 25)( 24, 26)
( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 37, 44)( 38, 43)( 39, 42)( 40, 41)
( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)( 60, 62)
( 65,128)( 66,127)( 67,126)( 68,125)( 69,124)( 70,123)( 71,122)( 72,121)
( 73,120)( 74,119)( 75,118)( 76,117)( 77,116)( 78,115)( 79,114)( 80,113)
( 81,104)( 82,103)( 83,102)( 84,101)( 85,100)( 86, 99)( 87, 98)( 88, 97)
( 89,112)( 90,111)( 91,110)( 92,109)( 93,108)( 94,107)( 95,106)( 96,105);;
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,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope