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## G = C42⋊2D5order 160 = 25·5

### 2nd semidirect product of C42 and D5 acting via D5/C5=C2

Aliases: C422D5, (C4×C20)⋊1C2, (C2×C4).63D10, C51(C422C2), C2.8(C4○D20), C10.6(C4○D4), C10.D41C2, D10⋊C4.1C2, (C2×C20).75C22, (C2×C10).17C23, (C2×Dic5).4C22, (C22×D5).3C22, C22.38(C22×D5), SmallGroup(160,97)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊2D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — D10⋊C4 — C42⋊2D5
 Lower central C5 — C2×C10 — C42⋊2D5
 Upper central C1 — C22 — C42

Generators and relations for C422D5
G = < a,b,c,d | a4=b4=c5=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >

Subgroups: 192 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C422C2, C2×Dic5, C2×C20, C22×D5, C10.D4, D10⋊C4, C4×C20, C422D5
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C422C2, C22×D5, C4○D20, C422D5

Smallest permutation representation of C422D5
On 80 points
Generators in S80
```(1 54 14 44)(2 55 15 45)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)```

`G:=sub<Sym(80)| (1,54,14,44)(2,55,15,45)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77)>;`

`G:=Group( (1,54,14,44)(2,55,15,45)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77) );`

`G=PermutationGroup([[(1,54,14,44),(2,55,15,45),(3,51,11,41),(4,52,12,42),(5,53,13,43),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70)], [(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(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),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,71),(62,75),(63,74),(64,73),(65,72),(66,76),(67,80),(68,79),(69,78),(70,77)]])`

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 D5 C4○D4 D10 C4○D20 kernel C42⋊2D5 C10.D4 D10⋊C4 C4×C20 C42 C10 C2×C4 C2 # reps 1 3 3 1 2 6 6 24

Matrix representation of C422D5 in GL4(𝔽41) generated by

 15 5 0 0 4 26 0 0 0 0 9 0 0 0 0 9
,
 9 0 0 0 0 9 0 0 0 0 17 40 0 0 1 24
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 40 34
,
 1 13 0 0 0 40 0 0 0 0 1 0 0 0 34 40
`G:=sub<GL(4,GF(41))| [15,4,0,0,5,26,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,17,1,0,0,40,24],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,13,40,0,0,0,0,1,34,0,0,0,40] >;`

C422D5 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2D_5`
`% in TeX`

`G:=Group("C4^2:2D5");`
`// GroupNames label`

`G:=SmallGroup(160,97);`
`// by ID`

`G=gap.SmallGroup(160,97);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,55,506,86,4613]);`
`// Polycyclic`

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

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