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# Polytope of Type {3,4,10}

Atlas Canonical Name : {3,4,10}*480
Also Known As : {{3,4},{4,10|2}}. if this polytope has another name.
Group : SmallGroup(480,1193)
Rank : 4
Schlafli Type : {3,4,10}
Number of vertices, edges, etc : 6, 12, 40, 10
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,4,10,2} of size 960
{3,4,10,4} of size 1920
Vertex Figure Of :
{2,3,4,10} of size 960
{3,3,4,10} of size 1920
{4,3,4,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
4-fold quotients : {3,2,10}*120
5-fold quotients : {3,4,2}*96
8-fold quotients : {3,2,5}*60
10-fold quotients : {3,4,2}*48
20-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,4,20}*960, {3,8,10}*960, {6,4,10}*960
3-fold covers : {9,4,10}*1440, {3,12,10}*1440, {3,4,30}*1440
4-fold covers : {3,8,20}*1920, {3,4,20}*1920, {3,8,10}*1920, {3,4,40}*1920, {12,4,10}*1920b, {6,4,20}*1920b, {6,4,10}*1920, {12,4,10}*1920c, {6,8,10}*1920a, {6,8,10}*1920b
Permutation Representation (GAP) :
```s0 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120);;
s1 := (  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 25)(  6, 26)(  7, 28)(  8, 27)
(  9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)( 16, 35)
( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)( 55, 56)
( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)( 67, 88)
( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)(111,112)
(115,116)(119,120);;
s2 := (  1,  4)(  2,  3)(  5, 20)(  6, 19)(  7, 18)(  8, 17)(  9, 16)( 10, 15)
( 11, 14)( 12, 13)( 21, 24)( 22, 23)( 25, 40)( 26, 39)( 27, 38)( 28, 37)
( 29, 36)( 30, 35)( 31, 34)( 32, 33)( 41, 44)( 42, 43)( 45, 60)( 46, 59)
( 47, 58)( 48, 57)( 49, 56)( 50, 55)( 51, 54)( 52, 53)( 61, 64)( 62, 63)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89, 96)( 90, 95)
( 91, 94)( 92, 93)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)
(109,116)(110,115)(111,114)(112,113);;
s3 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 61)(  6, 62)(  7, 63)(  8, 64)
(  9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 73)( 14, 74)( 15, 75)( 16, 76)
( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 85)( 22, 86)( 23, 87)( 24, 88)
( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 97)( 30, 98)( 31, 99)( 32,100)
( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)( 40, 92)
( 41,105)( 42,106)( 43,107)( 44,108)( 45,101)( 46,102)( 47,103)( 48,104)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,109)( 58,110)( 59,111)( 60,112);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(120)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)
( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)
( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)
( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)
( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)
( 98,119)( 99,118)(100,120);
s1 := Sym(120)!(  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 25)(  6, 26)(  7, 28)
(  8, 27)(  9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)
( 55, 56)( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)
( 67, 88)( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)
( 75, 96)( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)
(111,112)(115,116)(119,120);
s2 := Sym(120)!(  1,  4)(  2,  3)(  5, 20)(  6, 19)(  7, 18)(  8, 17)(  9, 16)
( 10, 15)( 11, 14)( 12, 13)( 21, 24)( 22, 23)( 25, 40)( 26, 39)( 27, 38)
( 28, 37)( 29, 36)( 30, 35)( 31, 34)( 32, 33)( 41, 44)( 42, 43)( 45, 60)
( 46, 59)( 47, 58)( 48, 57)( 49, 56)( 50, 55)( 51, 54)( 52, 53)( 61, 64)
( 62, 63)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)
( 72, 73)( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89, 96)
( 90, 95)( 91, 94)( 92, 93)(101,104)(102,103)(105,120)(106,119)(107,118)
(108,117)(109,116)(110,115)(111,114)(112,113);
s3 := Sym(120)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 61)(  6, 62)(  7, 63)
(  8, 64)(  9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 73)( 14, 74)( 15, 75)
( 16, 76)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 85)( 22, 86)( 23, 87)
( 24, 88)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 97)( 30, 98)( 31, 99)
( 32,100)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)
( 40, 92)( 41,105)( 42,106)( 43,107)( 44,108)( 45,101)( 46,102)( 47,103)
( 48,104)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)
( 56,116)( 57,109)( 58,110)( 59,111)( 60,112);
poly := sub<Sym(120)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope