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## G = 2+ 1+4⋊2D5order 320 = 26·5

### 2nd semidirect product of 2+ 1+4 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — 2+ 1+4⋊2D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C23⋊D10 — 2+ 1+4⋊2D5
 Lower central C5 — C10 — C22×C10 — 2+ 1+4⋊2D5
 Upper central C1 — C2 — C23 — 2+ 1+4

Generators and relations for 2+ 1+42D5
G = < a,b,c,d,e,f | a4=b2=d2=e5=f2=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, faf=a-1cd, fcf=bc=cb, fdf=bd=db, be=eb, bf=fb, dcd=a2c, ce=ec, de=ed, fef=e-1 >

Subgroups: 910 in 198 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C23⋊C4, C22≀C2, 2+ 1+4, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C22×C10, C22×C10, C2≀C22, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, D4×C10, C5×C4○D4, C23×D5, C23⋊Dic5, C23⋊D10, C5×2+ 1+4, 2+ 1+42D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, C2≀C22, C2×C5⋊D4, C242D5, 2+ 1+42D5

Smallest permutation representation of 2+ 1+42D5
On 40 points
Generators in S40
```(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 34 9 39)(2 35 10 40)(3 31 6 36)(4 32 7 37)(5 33 8 38)(11 21 16 26)(12 22 17 27)(13 23 18 28)(14 24 19 29)(15 25 20 30)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(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)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 5A 5B 10A 10B 10C ··· 10T 20A ··· 20L order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 10 10 10 ··· 10 20 ··· 20 size 1 1 2 2 2 4 4 4 20 20 4 4 4 40 40 40 2 2 2 2 4 ··· 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 8 type + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D5 D10 C5⋊D4 C5⋊D4 C2≀C22 2+ 1+4⋊2D5 kernel 2+ 1+4⋊2D5 C23⋊Dic5 C23⋊D10 C5×2+ 1+4 C2×C20 C22×C10 2+ 1+4 C2×D4 C2×C4 C23 C5 C1 # reps 1 3 3 1 3 3 2 6 12 12 2 2

Matrix representation of 2+ 1+42D5 in GL6(𝔽41)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40 0 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 3 7 0 0 0 0 7 3 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 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0

`G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[3,7,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;`

2+ 1+42D5 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes_2D_5`
`% in TeX`

`G:=Group("ES+(2,2):2D5");`
`// GroupNames label`

`G:=SmallGroup(320,871);`
`// by ID`

`G=gap.SmallGroup(320,871);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,570,438,12550]);`
`// Polycyclic`

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

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