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## G = D20⋊5D4order 320 = 26·5

### 5th semidirect product of D20 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊5D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4⋊6D10 — D20⋊5D4
 Lower central C5 — C10 — C2×C20 — D20⋊5D4
 Upper central C1 — C2 — C2×C4 — C4⋊1D4

Generators and relations for D205D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a5b, dbd=a10b, dcd=c-1 >

Subgroups: 750 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C52C8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, D44D4, C4.Dic5, D4⋊D5, D4.D5, C4×C20, C4○D20, D4×D5, D42D5, C2×C5⋊D4, D4×C10, D4×C10, D204C4, C20.D4, D4.D10, C5×C41D4, D46D10, D205D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D44D4, D4×D5, C2×C5⋊D4, C23⋊D10, D205D4

Smallest permutation representation of D205D4
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 2 4 4 8 20 20 2 2 4 4 20 20 2 2 40 40 2 ··· 2 8 ··· 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 C5⋊D4 D4⋊4D4 D4×D5 D20⋊5D4 kernel D20⋊5D4 D20⋊4C4 C20.D4 D4.D10 C5×C4⋊1D4 D4⋊6D10 Dic10 D20 C22×C10 C4⋊1D4 C42 C2×D4 C23 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 2 4 8 2 4 8

Matrix representation of D205D4 in GL4(𝔽41) generated by

 23 36 0 0 23 18 0 0 23 37 0 25 22 37 16 0
,
 1 0 9 0 1 0 25 25 0 0 40 0 18 23 40 0
,
 1 39 0 0 1 40 0 0 18 23 40 0 18 23 0 40
,
 1 39 0 0 0 40 0 0 0 23 0 1 0 23 1 0
`G:=sub<GL(4,GF(41))| [23,23,23,22,36,18,37,37,0,0,0,16,0,0,25,0],[1,1,0,18,0,0,0,23,9,25,40,40,0,25,0,0],[1,1,18,18,39,40,23,23,0,0,40,0,0,0,0,40],[1,0,0,0,39,40,23,23,0,0,0,1,0,0,1,0] >;`

D205D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_5D_4`
`% in TeX`

`G:=Group("D20:5D4");`
`// GroupNames label`

`G:=SmallGroup(320,704);`
`// by ID`

`G=gap.SmallGroup(320,704);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,1123,570,297,136,1684,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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