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# Polytope of Type {10,4,8}

Atlas Canonical Name : {10,4,8}*1280b
if this polytope has a name.
Group : SmallGroup(1280,323566)
Rank : 4
Schlafli Type : {10,4,8}
Number of vertices, edges, etc : 10, 40, 32, 16
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 : {10,4,4}*640
4-fold quotients : {10,4,4}*320
5-fold quotients : {2,4,8}*256b
8-fold quotients : {10,2,4}*160, {10,4,2}*160
10-fold quotients : {2,4,4}*128
16-fold quotients : {5,2,4}*80, {10,2,2}*80
20-fold quotients : {2,4,4}*64
32-fold quotients : {5,2,2}*40
40-fold quotients : {2,2,4}*32, {2,4,2}*32
80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 41)(  2, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)(  8, 49)
(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 56)
( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)( 24, 63)
( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)( 32, 75)
( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)
( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)
( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)
( 97,140)( 98,139)( 99,138)(100,137)(101,141)(102,145)(103,144)(104,143)
(105,142)(106,146)(107,150)(108,149)(109,148)(110,147)(111,151)(112,155)
(113,154)(114,153)(115,152)(116,156)(117,160)(118,159)(119,158)(120,157);;
s1 := (  1, 43)(  2, 42)(  3, 41)(  4, 45)(  5, 44)(  6, 48)(  7, 47)(  8, 46)
(  9, 50)( 10, 49)( 11, 53)( 12, 52)( 13, 51)( 14, 55)( 15, 54)( 16, 58)
( 17, 57)( 18, 56)( 19, 60)( 20, 59)( 21, 78)( 22, 77)( 23, 76)( 24, 80)
( 25, 79)( 26, 73)( 27, 72)( 28, 71)( 29, 75)( 30, 74)( 31, 68)( 32, 67)
( 33, 66)( 34, 70)( 35, 69)( 36, 63)( 37, 62)( 38, 61)( 39, 65)( 40, 64)
( 81,123)( 82,122)( 83,121)( 84,125)( 85,124)( 86,128)( 87,127)( 88,126)
( 89,130)( 90,129)( 91,133)( 92,132)( 93,131)( 94,135)( 95,134)( 96,138)
( 97,137)( 98,136)( 99,140)(100,139)(101,158)(102,157)(103,156)(104,160)
(105,159)(106,153)(107,152)(108,151)(109,155)(110,154)(111,148)(112,147)
(113,146)(114,150)(115,149)(116,143)(117,142)(118,141)(119,145)(120,144);;
s2 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 71, 76)
( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81,101)( 82,102)( 83,103)( 84,104)
( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,116)( 92,117)
( 93,118)( 94,119)( 95,120)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)
(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)
(129,149)(130,150)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)
(137,152)(138,153)(139,154)(140,155);;
s3 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,116)( 22,117)( 23,118)( 24,119)
( 25,120)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,101)( 37,102)( 38,103)( 39,104)( 40,105)
( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,156)( 62,157)( 63,158)( 64,159)
( 65,160)( 66,151)( 67,152)( 68,153)( 69,154)( 70,155)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 76,141)( 77,142)( 78,143)( 79,144)( 80,145);;
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
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(160)!(  1, 41)(  2, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)
(  8, 49)(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)
( 16, 56)( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)
( 24, 63)( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)
( 32, 75)( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)
( 40, 77)( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)
( 88,129)( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)
( 96,136)( 97,140)( 98,139)( 99,138)(100,137)(101,141)(102,145)(103,144)
(104,143)(105,142)(106,146)(107,150)(108,149)(109,148)(110,147)(111,151)
(112,155)(113,154)(114,153)(115,152)(116,156)(117,160)(118,159)(119,158)
(120,157);
s1 := Sym(160)!(  1, 43)(  2, 42)(  3, 41)(  4, 45)(  5, 44)(  6, 48)(  7, 47)
(  8, 46)(  9, 50)( 10, 49)( 11, 53)( 12, 52)( 13, 51)( 14, 55)( 15, 54)
( 16, 58)( 17, 57)( 18, 56)( 19, 60)( 20, 59)( 21, 78)( 22, 77)( 23, 76)
( 24, 80)( 25, 79)( 26, 73)( 27, 72)( 28, 71)( 29, 75)( 30, 74)( 31, 68)
( 32, 67)( 33, 66)( 34, 70)( 35, 69)( 36, 63)( 37, 62)( 38, 61)( 39, 65)
( 40, 64)( 81,123)( 82,122)( 83,121)( 84,125)( 85,124)( 86,128)( 87,127)
( 88,126)( 89,130)( 90,129)( 91,133)( 92,132)( 93,131)( 94,135)( 95,134)
( 96,138)( 97,137)( 98,136)( 99,140)(100,139)(101,158)(102,157)(103,156)
(104,160)(105,159)(106,153)(107,152)(108,151)(109,155)(110,154)(111,148)
(112,147)(113,146)(114,150)(115,149)(116,143)(117,142)(118,141)(119,145)
(120,144);
s2 := Sym(160)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)
( 71, 76)( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81,101)( 82,102)( 83,103)
( 84,104)( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,116)
( 92,117)( 93,118)( 94,119)( 95,120)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)
(128,148)(129,149)(130,150)(131,156)(132,157)(133,158)(134,159)(135,160)
(136,151)(137,152)(138,153)(139,154)(140,155);
s3 := Sym(160)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,116)( 22,117)( 23,118)
( 24,119)( 25,120)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,101)( 37,102)( 38,103)( 39,104)
( 40,105)( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)
( 48,128)( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)
( 56,136)( 57,137)( 58,138)( 59,139)( 60,140)( 61,156)( 62,157)( 63,158)
( 64,159)( 65,160)( 66,151)( 67,152)( 68,153)( 69,154)( 70,155)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 76,141)( 77,142)( 78,143)( 79,144)
( 80,145);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
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