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## G = D10.S4order 480 = 25·3·5

### 3rd non-split extension by D10 of S4 acting via S4/A4=C2

Aliases: D10.3S4, D5.GL2(𝔽3), SL2(𝔽3)⋊2F5, D5.CSU2(𝔽3), Q8⋊(C3⋊F5), C5⋊(Q8⋊Dic3), (C5×Q8)⋊Dic3, (Q8×D5).2S3, C2.3(A4⋊F5), C10.2(A4⋊C4), (C5×SL2(𝔽3))⋊2C4, (D5×SL2(𝔽3)).2C2, SmallGroup(480,962)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — D10.S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D10.S4
 Lower central C5×SL2(𝔽3) — D10.S4
 Upper central C1 — C2

Generators and relations for D10.S4
G = < a,b,c,d,e,f | a10=b2=e3=1, c2=d2=a5, f2=a-1b, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a7, bc=cb, bd=db, be=eb, fbf-1=a6b, dcd-1=a5c, ece-1=a5cd, fcf-1=cd, ede-1=c, fdf-1=a5d, fef-1=e-1 >

Character table of D10.S4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6A 6B 6C 8A 8B 8C 8D 10 15A 15B 20 30A 30B size 1 1 5 5 8 6 30 60 60 4 8 40 40 30 30 30 30 4 16 16 24 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 -1 -i i 1 1 -1 -1 -i i -i i 1 1 1 1 1 1 linear of order 4 ρ4 1 1 -1 -1 1 1 -1 i -i 1 1 -1 -1 i -i i -i 1 1 1 1 1 1 linear of order 4 ρ5 2 2 2 2 -1 2 2 0 0 2 -1 -1 -1 0 0 0 0 2 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 -2 -1 2 -2 0 0 2 -1 1 1 0 0 0 0 2 -1 -1 2 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ7 2 -2 -2 2 -1 0 0 0 0 2 1 -1 1 -√2 -√2 √2 √2 -2 -1 -1 0 1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ8 2 -2 -2 2 -1 0 0 0 0 2 1 -1 1 √2 √2 -√2 -√2 -2 -1 -1 0 1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ9 2 -2 2 -2 -1 0 0 0 0 2 1 1 -1 √-2 -√-2 -√-2 √-2 -2 -1 -1 0 1 1 complex lifted from GL2(𝔽3) ρ10 2 -2 2 -2 -1 0 0 0 0 2 1 1 -1 -√-2 √-2 √-2 -√-2 -2 -1 -1 0 1 1 complex lifted from GL2(𝔽3) ρ11 3 3 3 3 0 -1 -1 1 1 3 0 0 0 -1 -1 -1 -1 3 0 0 -1 0 0 orthogonal lifted from S4 ρ12 3 3 3 3 0 -1 -1 -1 -1 3 0 0 0 1 1 1 1 3 0 0 -1 0 0 orthogonal lifted from S4 ρ13 3 3 -3 -3 0 -1 1 i -i 3 0 0 0 -i i -i i 3 0 0 -1 0 0 complex lifted from A4⋊C4 ρ14 3 3 -3 -3 0 -1 1 -i i 3 0 0 0 i -i i -i 3 0 0 -1 0 0 complex lifted from A4⋊C4 ρ15 4 -4 4 -4 1 0 0 0 0 4 -1 -1 1 0 0 0 0 -4 1 1 0 -1 -1 orthogonal lifted from GL2(𝔽3) ρ16 4 4 0 0 4 4 0 0 0 -1 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 -4 -4 4 1 0 0 0 0 4 -1 1 -1 0 0 0 0 -4 1 1 0 -1 -1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ18 4 4 0 0 -2 4 0 0 0 -1 -2 0 0 0 0 0 0 -1 1-√-15/2 1+√-15/2 -1 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ19 4 4 0 0 -2 4 0 0 0 -1 -2 0 0 0 0 0 0 -1 1+√-15/2 1-√-15/2 -1 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ20 8 -8 0 0 -4 0 0 0 0 -2 4 0 0 0 0 0 0 2 1 1 0 -1 -1 symplectic faithful, Schur index 2 ρ21 8 -8 0 0 2 0 0 0 0 -2 -2 0 0 0 0 0 0 2 -1+√-15/2 -1-√-15/2 0 1+√-15/2 1-√-15/2 complex faithful ρ22 8 -8 0 0 2 0 0 0 0 -2 -2 0 0 0 0 0 0 2 -1-√-15/2 -1+√-15/2 0 1-√-15/2 1+√-15/2 complex faithful ρ23 12 12 0 0 0 -4 0 0 0 -3 0 0 0 0 0 0 0 -3 0 0 1 0 0 orthogonal lifted from A4⋊F5

Smallest permutation representation of D10.S4
On 40 points
Generators in S40
```(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 26)(22 25)(23 24)(27 30)(28 29)(31 38)(32 37)(33 36)(34 35)(39 40)
(1 40 6 35)(2 31 7 36)(3 32 8 37)(4 33 9 38)(5 34 10 39)(11 24 16 29)(12 25 17 30)(13 26 18 21)(14 27 19 22)(15 28 20 23)
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 37 26 32)(22 38 27 33)(23 39 28 34)(24 40 29 35)(25 31 30 36)
(11 24 40)(12 25 31)(13 26 32)(14 27 33)(15 28 34)(16 29 35)(17 30 36)(18 21 37)(19 22 38)(20 23 39)
(1 6)(2 9 10 3)(4 5 8 7)(12 14 20 18)(13 17 19 15)(21 31 27 39)(22 34 26 36)(23 37 25 33)(24 40)(28 32 30 38)(29 35)```

`G:=sub<Sym(40)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40), (1,40,6,35)(2,31,7,36)(3,32,8,37)(4,33,9,38)(5,34,10,39)(11,24,16,29)(12,25,17,30)(13,26,18,21)(14,27,19,22)(15,28,20,23), (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20)(21,37,26,32)(22,38,27,33)(23,39,28,34)(24,40,29,35)(25,31,30,36), (11,24,40)(12,25,31)(13,26,32)(14,27,33)(15,28,34)(16,29,35)(17,30,36)(18,21,37)(19,22,38)(20,23,39), (1,6)(2,9,10,3)(4,5,8,7)(12,14,20,18)(13,17,19,15)(21,31,27,39)(22,34,26,36)(23,37,25,33)(24,40)(28,32,30,38)(29,35)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40), (1,40,6,35)(2,31,7,36)(3,32,8,37)(4,33,9,38)(5,34,10,39)(11,24,16,29)(12,25,17,30)(13,26,18,21)(14,27,19,22)(15,28,20,23), (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20)(21,37,26,32)(22,38,27,33)(23,39,28,34)(24,40,29,35)(25,31,30,36), (11,24,40)(12,25,31)(13,26,32)(14,27,33)(15,28,34)(16,29,35)(17,30,36)(18,21,37)(19,22,38)(20,23,39), (1,6)(2,9,10,3)(4,5,8,7)(12,14,20,18)(13,17,19,15)(21,31,27,39)(22,34,26,36)(23,37,25,33)(24,40)(28,32,30,38)(29,35) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,26),(22,25),(23,24),(27,30),(28,29),(31,38),(32,37),(33,36),(34,35),(39,40)], [(1,40,6,35),(2,31,7,36),(3,32,8,37),(4,33,9,38),(5,34,10,39),(11,24,16,29),(12,25,17,30),(13,26,18,21),(14,27,19,22),(15,28,20,23)], [(1,11,6,16),(2,12,7,17),(3,13,8,18),(4,14,9,19),(5,15,10,20),(21,37,26,32),(22,38,27,33),(23,39,28,34),(24,40,29,35),(25,31,30,36)], [(11,24,40),(12,25,31),(13,26,32),(14,27,33),(15,28,34),(16,29,35),(17,30,36),(18,21,37),(19,22,38),(20,23,39)], [(1,6),(2,9,10,3),(4,5,8,7),(12,14,20,18),(13,17,19,15),(21,31,27,39),(22,34,26,36),(23,37,25,33),(24,40),(28,32,30,38),(29,35)]])`

Matrix representation of D10.S4 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 240 239 4 0 0 0 0 240 0 0 0 1 0 240 0 0 0 0 0 59 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 240 0 0 0 1 0 240 0 0 0 0 0 240 0 0 0 0 0 59 1
,
 68 67 0 0 0 0 136 173 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 106 67 0 0 0 0 174 135 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 177 0 0 0 0 177 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 4 0 0 1 0 0 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,1,0,0,0,240,0,0,0,0,0,239,240,240,59,0,0,4,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,240,240,240,59,0,0,0,0,0,1],[68,136,0,0,0,0,67,173,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[106,174,0,0,0,0,67,135,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,240,1,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,4,0,1] >;`

D10.S4 in GAP, Magma, Sage, TeX

`D_{10}.S_4`
`% in TeX`

`G:=Group("D10.S4");`
`// GroupNames label`

`G:=SmallGroup(480,962);`
`// by ID`

`G=gap.SmallGroup(480,962);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,14,170,1011,682,4204,3168,172,2525,1909,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^10=b^2=e^3=1,c^2=d^2=a^5,f^2=a^-1*b,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^7,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^6*b,d*c*d^-1=a^5*c,e*c*e^-1=a^5*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=a^5*d,f*e*f^-1=e^-1>;`
`// generators/relations`

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