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## G = C24⋊2D5order 160 = 25·5

### 1st semidirect product of C24 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24⋊2D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C24⋊2D5
 Lower central C5 — C2×C10 — C24⋊2D5
 Upper central C1 — C22 — C24

Generators and relations for C242D5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 368 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×C10, C22≀C2, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C23.D5, C2×C5⋊D4, C23×C10, C242D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, C2×C5⋊D4, C242D5

Smallest permutation representation of C242D5
On 40 points
Generators in S40
```(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(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 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 25)(2 24)(3 23)(4 22)(5 21)(6 28)(7 27)(8 26)(9 30)(10 29)(11 33)(12 32)(13 31)(14 35)(15 34)(16 38)(17 37)(18 36)(19 40)(20 39)```

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 5A 5B 10A ··· 10AD order 1 2 2 2 2 ··· 2 2 4 4 4 5 5 10 ··· 10 size 1 1 1 1 2 ··· 2 20 20 20 20 2 2 2 ··· 2

46 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D5 D10 C5⋊D4 kernel C24⋊2D5 C23.D5 C2×C5⋊D4 C23×C10 C2×C10 C24 C23 C22 # reps 1 3 3 1 6 2 6 24

Matrix representation of C242D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 16 0 0 0 0 18 0 0 0 0 10 0 0 0 0 37
,
 0 18 0 0 16 0 0 0 0 0 0 37 0 0 10 0
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,0,0,0,0,18,0,0,0,0,10,0,0,0,0,37],[0,16,0,0,18,0,0,0,0,0,0,10,0,0,37,0] >;`

C242D5 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_2D_5`
`% in TeX`

`G:=Group("C2^4:2D5");`
`// GroupNames label`

`G:=SmallGroup(160,174);`
`// by ID`

`G=gap.SmallGroup(160,174);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,4613]);`
`// Polycyclic`

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

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