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## G = D5⋊(C4.D4)  order 320 = 26·5

### The semidirect product of D5 and C4.D4 acting via C4.D4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5⋊(C4.D4)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — D5⋊M4(2) — D5⋊(C4.D4)
 Lower central C5 — C10 — C2×C10 — D5⋊(C4.D4)
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for D5⋊(C4.D4)
G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c, bab=cac-1=a-1, dad-1=eae-1=a3, cbc-1=a3b, dbd-1=ebe-1=a2b, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 970 in 186 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C4.D4, C2×M4(2), C22×D4, C5⋊C8, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4.D4, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C23.F5, D5⋊M4(2), C2×D4×D5, D5⋊(C4.D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C4.D4, C2×C22⋊C4, C2×F5, C2×C4.D4, C22⋊F5, C22×F5, C2×C22⋊F5, D5⋊(C4.D4)

Smallest permutation representation of D5⋊(C4.D4)
On 40 points
Generators in S40
```(1 26 37 24 15)(2 17 27 16 38)(3 9 18 39 28)(4 40 10 29 19)(5 30 33 20 11)(6 21 31 12 34)(7 13 22 35 32)(8 36 14 25 23)
(1 11)(2 34)(3 32)(4 23)(5 15)(6 38)(7 28)(8 19)(9 35)(10 14)(12 17)(13 39)(16 21)(18 22)(20 26)(24 30)(25 40)(27 31)(29 36)(33 37)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)
(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 6 3 4 5 2 7 8)(9 10 11 16 13 14 15 12)(17 22 23 24 21 18 19 20)(25 26 31 28 29 30 27 32)(33 38 35 36 37 34 39 40)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 F5 C4.D4 C2×F5 C2×F5 C22⋊F5 D5⋊(C4.D4) kernel D5⋊(C4.D4) C23.F5 D5⋊M4(2) C2×D4×D5 C2×D20 D4×C10 C23×D5 C4×D5 C2×D4 D5 C2×C4 C23 C4 C1 # reps 1 4 2 1 2 2 4 4 1 2 1 2 4 2

Matrix representation of D5⋊(C4.D4) in GL8(ℤ)

 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 -1 -1 -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 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0
,
 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 -1 0 -1 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 -1 0 -1 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0

`G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[1,1,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[1,1,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

D5⋊(C4.D4) in GAP, Magma, Sage, TeX

`D_5\rtimes (C_4.D_4)`
`% in TeX`

`G:=Group("D5:(C4.D4)");`
`// GroupNames label`

`G:=SmallGroup(320,1116);`
`// by ID`

`G=gap.SmallGroup(320,1116);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,297,136,1684,6278,1595]);`
`// Polycyclic`

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

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