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## G = C4⋊C4⋊5F5order 320 = 26·5

### 3rd semidirect product of C4⋊C4 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4⋊5F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4×F5 — C4⋊C4⋊5F5
 Lower central C5 — C2×C10 — C4⋊C4⋊5F5
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C45F5
G = < a,b,c,d | a4=b4=c5=d4=1, bab-1=a-1, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 618 in 154 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C42, C4⋊C4, C4⋊C4, C22×C4, Dic5, C20, F5, D10, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C23.63C23, C10.D4, C4⋊Dic5, C5×C4⋊C4, C4×F5, C4⋊F5, C2×C4×D5, C22×F5, D10.3Q8, D5×C4⋊C4, C2×C4×F5, C2×C4⋊F5, C4⋊C45F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, F5, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×F5, C23.63C23, C22×F5, D10.C23, D4×F5, Q8×F5, C4⋊C45F5

Smallest permutation representation of C4⋊C45F5
On 80 points
Generators in S80
```(1 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 46 76)(42 72 47 77)(43 73 48 78)(44 74 49 79)(45 75 50 80)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(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 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 21 16 36)(2 23 20 39)(3 25 19 37)(4 22 18 40)(5 24 17 38)(6 26 11 31)(7 28 15 34)(8 30 14 32)(9 27 13 35)(10 29 12 33)(41 61 56 76)(42 63 60 79)(43 65 59 77)(44 62 58 80)(45 64 57 78)(46 66 51 71)(47 68 55 74)(48 70 54 72)(49 67 53 75)(50 69 52 73)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 8 8 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 C4○D4 F5 C2×F5 D10.C23 D4×F5 Q8×F5 kernel C4⋊C4⋊5F5 D10.3Q8 D5×C4⋊C4 C2×C4×F5 C2×C4⋊F5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C2×F5 C2×F5 D10 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 4 1 1 1 4 2 2 2 2 8 1 3 4 1 1

Matrix representation of C4⋊C45F5 in GL8(𝔽41)

 1 2 0 0 0 0 0 0 40 40 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 12 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 32 23 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 34 10 0 0 0 0 0 0 28 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 32 23 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(41))| [1,40,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,32,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,23,9,0,0,0,0,0,0,0,0,34,28,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[32,0,0,0,0,0,0,0,23,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;`

C4⋊C45F5 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes_5F_5`
`% in TeX`

`G:=Group("C4:C4:5F5");`
`// GroupNames label`

`G:=SmallGroup(320,1049);`
`// by ID`

`G=gap.SmallGroup(320,1049);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,184,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

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