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## G = C4⋊C4⋊7D5order 160 = 25·5

### 1st semidirect product of C4⋊C4 and D5 acting through Inn(C4⋊C4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4⋊C4⋊7D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C4⋊C4⋊7D5
 Lower central C5 — C10 — C4⋊C4⋊7D5
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C47D5
G = < a,b,c,d | a4=b4=c5=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 216 in 76 conjugacy classes, 41 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C2×C4, C2×C4 [×2], C2×C4 [×7], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C42⋊C2, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C4⋊C47D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C4×D5 [×2], C22×D5, C2×C4×D5, D42D5, Q82D5, C4⋊C47D5

Smallest permutation representation of C4⋊C47D5
On 80 points
Generators in S80
```(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)```

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 2 ··· 2 5 5 5 5 10 10 10 10 2 2 2 ··· 2 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 C4×D5 D4⋊2D5 Q8⋊2D5 kernel C4⋊C4⋊7D5 C4×Dic5 C4⋊Dic5 D10⋊C4 C5×C4⋊C4 C2×C4×D5 C4×D5 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 2 1 2 1 1 8 2 4 6 8 2 2

Matrix representation of C4⋊C47D5 in GL5(𝔽41)

 1 0 0 0 0 0 32 0 0 0 0 0 9 0 0 0 0 0 40 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 1 0 0 0 40 0
,
 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 6 0 0 0 0 40

`G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,6,40] >;`

C4⋊C47D5 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes_7D_5`
`% in TeX`

`G:=Group("C4:C4:7D5");`
`// GroupNames label`

`G:=SmallGroup(160,113);`
`// by ID`

`G=gap.SmallGroup(160,113);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,188,50,4613]);`
`// Polycyclic`

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

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