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## G = D10.3Q8order 160 = 25·5

### 3rd non-split extension by D10 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.3Q8
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — D10.3Q8
 Lower central C5 — C10 — D10.3Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for D10.3Q8
G = < a,b,c,d | a10=b2=c4=1, d2=a4bc2, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=a5c-1 >

Subgroups: 268 in 76 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×6], C5, C2×C4, C2×C4 [×11], C23, D5 [×4], C10 [×3], C22×C4 [×3], Dic5, C20, F5 [×4], D10 [×2], D10 [×4], C2×C10, C2.C42, C4×D5 [×2], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C2×C4×D5, C22×F5 [×2], D10.3Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C2×F5, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8

Character table of D10.3Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 10A 10B 10C 20A 20B 20C 20D size 1 1 1 1 5 5 5 5 2 2 10 10 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 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 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 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 ρ4 1 1 1 1 1 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 ρ5 1 1 1 1 -1 -1 -1 -1 -1 -1 i 1 i i -i -i -i -i i 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 -1 -1 -i 1 -i -i i i i i -i 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 -1 -1 1 1 i -i -1 i i -i -i i -1 1 1 -i 1 -1 1 -1 -i -i i i linear of order 4 ρ8 1 1 -1 -1 -1 -1 1 1 i -i 1 i -i i i -i 1 -1 -1 -i 1 -1 1 -1 -i -i i i linear of order 4 ρ9 1 1 -1 -1 -1 -1 1 1 -i i -1 -i -i i i -i -1 1 1 i 1 -1 1 -1 i i -i -i linear of order 4 ρ10 1 1 -1 -1 -1 -1 1 1 -i i 1 -i i -i -i i 1 -1 -1 i 1 -1 1 -1 i i -i -i linear of order 4 ρ11 1 1 1 1 -1 -1 -1 -1 1 1 i -1 -i -i i i -i -i i -1 1 1 1 1 1 1 1 1 linear of order 4 ρ12 1 1 1 1 -1 -1 -1 -1 1 1 -i -1 i i -i -i i i -i -1 1 1 1 1 1 1 1 1 linear of order 4 ρ13 1 1 -1 -1 1 1 -1 -1 i -i i -i -1 1 -1 1 -i i -i i 1 -1 1 -1 -i -i i i linear of order 4 ρ14 1 1 -1 -1 1 1 -1 -1 i -i -i -i 1 -1 1 -1 i -i i i 1 -1 1 -1 -i -i i i linear of order 4 ρ15 1 1 -1 -1 1 1 -1 -1 -i i i i 1 -1 1 -1 -i i -i -i 1 -1 1 -1 i i -i -i linear of order 4 ρ16 1 1 -1 -1 1 1 -1 -1 -i i -i i -1 1 -1 1 i -i i -i 1 -1 1 -1 i i -i -i linear of order 4 ρ17 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 4 4 4 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ22 4 4 4 4 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 -√5 √5 √5 -√5 orthogonal lifted from C22⋊F5 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 √5 -√5 -√5 √5 orthogonal lifted from C22⋊F5 ρ25 4 4 -4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 -i -i i i complex lifted from C4×F5 ρ26 4 4 -4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 i i -i -i complex lifted from C4×F5 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 -√-5 √-5 -√-5 √-5 complex lifted from C4⋊F5 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 √-5 -√-5 √-5 -√-5 complex lifted from C4⋊F5

Smallest permutation representation of D10.3Q8
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 32)(12 31)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)
(1 33 30 11)(2 34 21 12)(3 35 22 13)(4 36 23 14)(5 37 24 15)(6 38 25 16)(7 39 26 17)(8 40 27 18)(9 31 28 19)(10 32 29 20)
(1 33 6 38)(2 40 5 31)(3 37 4 34)(7 35 10 36)(8 32 9 39)(11 25 16 30)(12 22 15 23)(13 29 14 26)(17 27 20 28)(18 24 19 21)```

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

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

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

Matrix representation of D10.3Q8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 1 0 0 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40 0 0 0
,
 1 2 0 0 0 0 0 40 0 0 0 0 0 0 7 27 0 14 0 0 0 34 27 14 0 0 14 27 34 0 0 0 14 0 27 7
,
 32 23 0 0 0 0 9 9 0 0 0 0 0 0 7 0 34 27 0 0 0 27 7 34 0 0 14 34 7 27 0 0 14 27 34 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[32,9,0,0,0,0,23,9,0,0,0,0,0,0,7,0,14,14,0,0,0,27,34,27,0,0,34,7,7,34,0,0,27,34,27,0] >;`

D10.3Q8 in GAP, Magma, Sage, TeX

`D_{10}._3Q_8`
`% in TeX`

`G:=Group("D10.3Q8");`
`// GroupNames label`

`G:=SmallGroup(160,81);`
`// by ID`

`G=gap.SmallGroup(160,81);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^4*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a^5*c^-1>;`
`// generators/relations`

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