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# Polytope of Type {4,20,4}

Atlas Canonical Name : {4,20,4}*1280a
if this polytope has a name.
Group : SmallGroup(1280,201150)
Rank : 4
Schlafli Type : {4,20,4}
Number of vertices, edges, etc : 4, 80, 80, 8
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
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 : {4,20,4}*640, {2,20,4}*640
4-fold quotients : {2,20,4}*320, {4,20,2}*320, {4,10,4}*320
5-fold quotients : {4,4,4}*256a
8-fold quotients : {2,20,2}*160, {2,10,4}*160, {4,10,2}*160
10-fold quotients : {4,4,4}*128, {2,4,4}*128
16-fold quotients : {2,10,2}*80
20-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
32-fold quotients : {2,5,2}*40
40-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 56)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 61)( 22, 62)( 23, 63)( 24, 64)
( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 69)( 30, 70)( 31, 71)( 32, 72)
( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)( 40, 80)
( 81,131)( 82,132)( 83,133)( 84,134)( 85,135)( 86,136)( 87,137)( 88,138)
( 89,139)( 90,140)( 91,121)( 92,122)( 93,123)( 94,124)( 95,125)( 96,126)
( 97,127)( 98,128)( 99,129)(100,130)(101,151)(102,152)(103,153)(104,154)
(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,141)(112,142)
(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150);;
s1 := (  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)(  8, 89)
(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)( 16, 96)
( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,101)( 22,105)( 23,104)( 24,103)
( 25,102)( 26,106)( 27,110)( 28,109)( 29,108)( 30,107)( 31,111)( 32,115)
( 33,114)( 34,113)( 35,112)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)
( 41,121)( 42,125)( 43,124)( 44,123)( 45,122)( 46,126)( 47,130)( 48,129)
( 49,128)( 50,127)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)( 56,136)
( 57,140)( 58,139)( 59,138)( 60,137)( 61,141)( 62,145)( 63,144)( 64,143)
( 65,142)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,151)( 72,155)
( 73,154)( 74,153)( 75,152)( 76,156)( 77,160)( 78,159)( 79,158)( 80,157);;
s2 := (  1,  3)(  4,  5)(  6,  8)(  9, 10)( 11, 13)( 14, 15)( 16, 18)( 19, 20)
( 21, 23)( 24, 25)( 26, 28)( 29, 30)( 31, 33)( 34, 35)( 36, 38)( 39, 40)
( 41, 43)( 44, 45)( 46, 48)( 49, 50)( 51, 53)( 54, 55)( 56, 58)( 59, 60)
( 61, 63)( 64, 65)( 66, 68)( 69, 70)( 71, 73)( 74, 75)( 76, 78)( 79, 80)
( 81,103)( 82,102)( 83,101)( 84,105)( 85,104)( 86,108)( 87,107)( 88,106)
( 89,110)( 90,109)( 91,113)( 92,112)( 93,111)( 94,115)( 95,114)( 96,118)
( 97,117)( 98,116)( 99,120)(100,119)(121,143)(122,142)(123,141)(124,145)
(125,144)(126,148)(127,147)(128,146)(129,150)(130,149)(131,153)(132,152)
(133,151)(134,155)(135,154)(136,158)(137,157)(138,156)(139,160)(140,159);;
s3 := (  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 56)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 66)( 22, 67)( 23, 68)( 24, 69)
( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)( 32, 77)
( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)
( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)( 88,128)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)( 96,136)
( 97,137)( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)
(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)
(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(160)!(  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)
( 16, 56)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 61)( 22, 62)( 23, 63)
( 24, 64)( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 69)( 30, 70)( 31, 71)
( 32, 72)( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)
( 40, 80)( 81,131)( 82,132)( 83,133)( 84,134)( 85,135)( 86,136)( 87,137)
( 88,138)( 89,139)( 90,140)( 91,121)( 92,122)( 93,123)( 94,124)( 95,125)
( 96,126)( 97,127)( 98,128)( 99,129)(100,130)(101,151)(102,152)(103,153)
(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,141)
(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)
(120,150);
s1 := Sym(160)!(  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)
(  8, 89)(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)
( 16, 96)( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,101)( 22,105)( 23,104)
( 24,103)( 25,102)( 26,106)( 27,110)( 28,109)( 29,108)( 30,107)( 31,111)
( 32,115)( 33,114)( 34,113)( 35,112)( 36,116)( 37,120)( 38,119)( 39,118)
( 40,117)( 41,121)( 42,125)( 43,124)( 44,123)( 45,122)( 46,126)( 47,130)
( 48,129)( 49,128)( 50,127)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)
( 56,136)( 57,140)( 58,139)( 59,138)( 60,137)( 61,141)( 62,145)( 63,144)
( 64,143)( 65,142)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,151)
( 72,155)( 73,154)( 74,153)( 75,152)( 76,156)( 77,160)( 78,159)( 79,158)
( 80,157);
s2 := Sym(160)!(  1,  3)(  4,  5)(  6,  8)(  9, 10)( 11, 13)( 14, 15)( 16, 18)
( 19, 20)( 21, 23)( 24, 25)( 26, 28)( 29, 30)( 31, 33)( 34, 35)( 36, 38)
( 39, 40)( 41, 43)( 44, 45)( 46, 48)( 49, 50)( 51, 53)( 54, 55)( 56, 58)
( 59, 60)( 61, 63)( 64, 65)( 66, 68)( 69, 70)( 71, 73)( 74, 75)( 76, 78)
( 79, 80)( 81,103)( 82,102)( 83,101)( 84,105)( 85,104)( 86,108)( 87,107)
( 88,106)( 89,110)( 90,109)( 91,113)( 92,112)( 93,111)( 94,115)( 95,114)
( 96,118)( 97,117)( 98,116)( 99,120)(100,119)(121,143)(122,142)(123,141)
(124,145)(125,144)(126,148)(127,147)(128,146)(129,150)(130,149)(131,153)
(132,152)(133,151)(134,155)(135,154)(136,158)(137,157)(138,156)(139,160)
(140,159);
s3 := Sym(160)!(  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)
( 16, 56)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 66)( 22, 67)( 23, 68)
( 24, 69)( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)
( 32, 77)( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)
( 40, 75)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)
( 88,128)( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)
( 96,136)( 97,137)( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)
(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)
(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)
(120,155);
poly := sub<Sym(160)|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,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope