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## G = D5⋊C4≀C2order 320 = 26·5

### The semidirect product of D5 and C4≀C2 acting via C4≀C2/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5⋊C4≀C2
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — C2×C4×F5 — D5⋊C4≀C2
 Lower central C5 — C10 — C20 — D5⋊C4≀C2
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D5⋊C4≀C2
G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=a3, bc=cb, bd=db, ebe-1=a2b, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 730 in 170 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42 [×3], C2×C8, M4(2) [×3], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10, C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C5⋊C8 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5 [×6], C22×D5, C22×D5, C2×C4≀C2, D5⋊C8, C4.F5 [×2], C4×F5 [×2], C4×F5, C22.F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C22×F5, D4⋊F5 [×2], Q82F5 [×2], D5⋊M4(2), C2×C4×F5, D5×C4○D4, D5⋊C4≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4≀C2 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4≀C2, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, D5⋊C4≀C2

Smallest permutation representation of D5⋊C4≀C2
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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 9)(2 6 5 7)(3 8 4 10)(11 18 12 20)(13 17 15 16)(14 19)(21 38 27 35)(22 40 26 33)(23 37 30 31)(24 39 29 34)(25 36 28 32)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4O 4P 5 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 20E order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 5 8 8 8 8 10 10 10 10 20 20 20 20 20 size 1 1 2 4 5 5 10 20 1 1 2 4 5 5 10 ··· 10 20 4 20 20 20 20 4 8 8 8 4 4 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 D4 C4≀C2 F5 C2×F5 C2×F5 C2×F5 C22⋊F5 C22⋊F5 D5⋊C4≀C2 kernel D5⋊C4≀C2 D4⋊F5 Q8⋊2F5 D5⋊M4(2) C2×C4×F5 D5×C4○D4 C4○D20 D4×D5 Q8×D5 C5×C4○D4 C4×D5 C2×Dic5 C22×D5 D5 C4○D4 C2×C4 D4 Q8 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 2 2 2 1 1 8 1 1 1 1 2 2 2

Matrix representation of D5⋊C4≀C2 in GL6(𝔽41)

 1 0 0 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 40 40 40 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1
,
 32 32 0 0 0 0 0 9 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
,
 9 9 0 0 0 0 23 32 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
,
 40 36 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1],[32,0,0,0,0,0,32,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,23,0,0,0,0,9,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,36,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;`

D5⋊C4≀C2 in GAP, Magma, Sage, TeX

`D_5\rtimes C_4\wr C_2`
`% in TeX`

`G:=Group("D5:C4wrC2");`
`// GroupNames label`

`G:=SmallGroup(320,1130);`
`// by ID`

`G=gap.SmallGroup(320,1130);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,438,102,6278,1595]);`
`// Polycyclic`

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

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