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## G = D40⋊C4order 320 = 26·5

### 2nd semidirect product of D40 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D40⋊C4
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — D4×F5 — D40⋊C4
 Lower central C5 — C10 — C20 — D40⋊C4
 Upper central C1 — C2 — C4 — D8

Generators and relations for D40⋊C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a32b >

Subgroups: 674 in 132 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×6], C4, C4 [×5], C22 [×9], C5, C8, C8 [×2], C2×C4 [×9], D4 [×2], D4 [×4], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8, D8 [×3], C22×C4 [×2], C2×D4 [×2], Dic5, C20, F5 [×4], D10, D10 [×6], C2×C10 [×2], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C52C8, C40, C5⋊C8, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5 [×2], C2×F5 [×6], C22×D5 [×2], D8⋊C4, C8×D5, D40, D4⋊D5 [×2], C5×D8, D5⋊C8, C4×F5, C4⋊F5 [×2], C22⋊F5 [×2], D4×D5 [×2], C22×F5 [×2], C8⋊F5, C40⋊C4, D20⋊C4 [×2], D5×D8, D4×F5 [×2], D40⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C8⋊C22 [×2], C2×F5 [×3], D8⋊C4, C22×F5, D4×F5, D40⋊C4

Character table of D40⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5 8A 8B 8C 8D 10A 10B 10C 20 40A 40B size 1 1 4 4 5 5 20 20 2 10 10 10 10 10 20 20 20 20 4 4 20 20 20 4 16 16 8 8 8 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 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 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 -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 -1 linear of order 2 ρ5 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 1 linear of order 2 ρ6 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 1 linear of order 2 ρ7 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 -1 linear of order 2 ρ8 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 -1 linear of order 2 ρ9 1 1 -1 1 -1 -1 1 -1 1 -1 i i -i -i i i -i -i 1 -1 i -i 1 1 -1 1 1 -1 -1 linear of order 4 ρ10 1 1 1 -1 -1 -1 -1 1 1 -1 i i -i -i -i -i i i 1 -1 i -i 1 1 1 -1 1 -1 -1 linear of order 4 ρ11 1 1 1 -1 -1 -1 -1 1 1 -1 -i -i i i i i -i -i 1 -1 -i i 1 1 1 -1 1 -1 -1 linear of order 4 ρ12 1 1 -1 1 -1 -1 1 -1 1 -1 -i -i i i -i -i i i 1 -1 -i i 1 1 -1 1 1 -1 -1 linear of order 4 ρ13 1 1 1 1 -1 -1 -1 -1 1 -1 i i -i -i -i i -i i 1 1 -i i -1 1 1 1 1 1 1 linear of order 4 ρ14 1 1 -1 -1 -1 -1 1 1 1 -1 i i -i -i i -i i -i 1 1 -i i -1 1 -1 -1 1 1 1 linear of order 4 ρ15 1 1 -1 -1 -1 -1 1 1 1 -1 -i -i i i -i i -i i 1 1 i -i -1 1 -1 -1 1 1 1 linear of order 4 ρ16 1 1 1 1 -1 -1 -1 -1 1 -1 -i -i i i i -i i -i 1 1 i -i -1 1 1 1 1 1 1 linear of order 4 ρ17 2 2 0 0 2 2 0 0 -2 -2 -2 2 -2 2 0 0 0 0 2 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ18 2 2 0 0 2 2 0 0 -2 -2 2 -2 2 -2 0 0 0 0 2 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ19 2 2 0 0 -2 -2 0 0 -2 2 -2i 2i 2i -2i 0 0 0 0 2 0 0 0 0 2 0 0 -2 0 0 complex lifted from C4○D4 ρ20 2 2 0 0 -2 -2 0 0 -2 2 2i -2i -2i 2i 0 0 0 0 2 0 0 0 0 2 0 0 -2 0 0 complex lifted from C4○D4 ρ21 4 4 -4 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 -1 4 0 0 0 -1 1 1 -1 -1 -1 orthogonal lifted from C2×F5 ρ22 4 -4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ23 4 4 4 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 -1 -4 0 0 0 -1 -1 1 -1 1 1 orthogonal lifted from C2×F5 ρ24 4 4 4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 -1 4 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 4 -4 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 4 4 -4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 -1 -4 0 0 0 -1 1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ27 8 8 0 0 0 0 0 0 -8 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4×F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 -√10 √10 orthogonal faithful ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 √10 -√10 orthogonal faithful

Smallest permutation representation of D40⋊C4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)
(1 21)(2 18 10 34)(3 15 19 7)(4 12 28 20)(5 9 37 33)(8 40 24 32)(11 31)(13 25 29 17)(14 22 38 30)(23 35 39 27)```

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

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

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

Matrix representation of D40⋊C4 in GL8(𝔽41)

 25 0 5 0 0 0 0 0 4 1 0 2 0 0 0 0 31 1 16 0 0 0 0 0 38 40 31 40 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 1 5 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 16 1 0 0 0 0 0 32 31 40 40 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 10 0 1 0 0 0 0 0 4 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40

`G:=sub<GL(8,GF(41))| [25,4,31,38,0,0,0,0,0,1,1,40,0,0,0,0,5,0,16,31,0,0,0,0,0,2,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0],[1,0,0,32,0,0,0,0,5,40,16,31,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0],[40,0,10,4,0,0,0,0,0,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40] >;`

D40⋊C4 in GAP, Magma, Sage, TeX

`D_{40}\rtimes C_4`
`% in TeX`

`G:=Group("D40:C4");`
`// GroupNames label`

`G:=SmallGroup(320,1069);`
`// by ID`

`G=gap.SmallGroup(320,1069);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,851,438,102,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^32*b>;`
`// generators/relations`

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