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## G = D10.D4order 160 = 25·5

### 1st non-split extension by D10 of D4 acting via D4/C2=C22

Aliases: D10.1D4, (C2×C4)⋊F5, (C2×C20)⋊1C4, C51(C23⋊C4), C22⋊F51C2, (C2×D20).2C2, (C22×D5)⋊2C4, C22.2(C2×F5), C2.4(C22⋊F5), C10.1(C22⋊C4), (C22×D5).14C22, (C2×C10).2(C2×C4), SmallGroup(160,74)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10.D4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C22⋊F5 — D10.D4
 Lower central C5 — C10 — C2×C10 — D10.D4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for D10.D4
G = < a,b,c,d | a10=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc-1 >

2C2
10C2
10C2
20C2
2C4
5C22
5C22
10C22
20C4
20C4
20C22
20C22
2D5
2D5
2C10
4D5
5C23
5C23
10C2×C4
10D4
10D4
10C2×C4
2C20
2D10
4D10
4F5
4F5
4D10
2D20
2D20

Character table of D10.D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 5 10A 10B 10C 20A 20B 20C 20D size 1 1 2 10 10 20 4 20 20 20 20 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 trivial ρ2 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 linear of order 2 ρ4 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 -i -i i i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 1 -1 -1 1 -1 i i -i -i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 -i i -i i 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 i -i i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 2 -2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 0 0 -4 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ12 4 -4 0 0 0 0 0 0 0 0 0 4 -4 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ13 4 4 4 0 0 0 4 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 0 0 0 0 0 -1 -1 1 1 √5 √5 -√5 -√5 orthogonal lifted from C22⋊F5 ρ15 4 4 -4 0 0 0 0 0 0 0 0 -1 -1 1 1 -√5 -√5 √5 √5 orthogonal lifted from C22⋊F5 ρ16 4 -4 0 0 0 0 0 0 0 0 0 -1 1 √5 -√5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 orthogonal faithful ρ17 4 -4 0 0 0 0 0 0 0 0 0 -1 1 -√5 √5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 orthogonal faithful ρ18 4 -4 0 0 0 0 0 0 0 0 0 -1 1 -√5 √5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 orthogonal faithful ρ19 4 -4 0 0 0 0 0 0 0 0 0 -1 1 √5 -√5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 orthogonal faithful

Smallest permutation representation of D10.D4
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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)
(1 24 12 36)(2 21 11 39)(3 28 20 32)(4 25 19 35)(5 22 18 38)(6 29 17 31)(7 26 16 34)(8 23 15 37)(9 30 14 40)(10 27 13 33)
(1 31 17 29)(2 38 16 22)(3 35 15 25)(4 32 14 28)(5 39 13 21)(6 36 12 24)(7 33 11 27)(8 40 20 30)(9 37 19 23)(10 34 18 26)```

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

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

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

D10.D4 is a maximal subgroup of
C42⋊F5  C422F5  (C2×D4)⋊F5  (C2×Q8)⋊F5  C23⋊F55C2  (C2×D4)⋊7F5  (C2×Q8)⋊7F5  D10.4D12  (C2×C60)⋊C4
D10.D4 is a maximal quotient of
C42⋊F5  C422F5  C42.F5  C42.2F5  C5⋊C2≀C4  C22⋊C4⋊F5  D10.1D8  D10.1Q16  (C2×C20)⋊1C8  C5⋊(C23⋊C8)  C22⋊F5⋊C4  D10.4D12  (C2×C60)⋊C4

Matrix representation of D10.D4 in GL4(𝔽41) generated by

 6 6 0 0 35 1 0 0 0 0 0 40 27 27 1 35
,
 25 16 0 0 2 16 0 0 32 27 28 39 14 30 2 13
,
 40 13 37 15 7 39 26 4 4 16 14 9 6 5 11 30
,
 34 0 39 0 1 1 0 39 5 27 7 0 30 20 40 40
`G:=sub<GL(4,GF(41))| [6,35,0,27,6,1,0,27,0,0,0,1,0,0,40,35],[25,2,32,14,16,16,27,30,0,0,28,2,0,0,39,13],[40,7,4,6,13,39,16,5,37,26,14,11,15,4,9,30],[34,1,5,30,0,1,27,20,39,0,7,40,0,39,0,40] >;`

D10.D4 in GAP, Magma, Sage, TeX

`D_{10}.D_4`
`% in TeX`

`G:=Group("D10.D4");`
`// GroupNames label`

`G:=SmallGroup(160,74);`
`// by ID`

`G=gap.SmallGroup(160,74);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,188,579,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^4*b*c^-1>;`
`// generators/relations`

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