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## G = C42⋊8F5order 320 = 26·5

### 5th semidirect product of C42 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊8F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4⋊F5 — C42⋊8F5
 Lower central C5 — C2×C10 — C42⋊8F5
 Upper central C1 — C22 — C42

Generators and relations for C428F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 714 in 186 conjugacy classes, 80 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×10], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×27], C23, D5, D5 [×3], C10 [×3], C42, C42 [×3], C4⋊C4 [×12], C22×C4 [×7], Dic5 [×6], C20 [×6], F5 [×4], D10 [×6], C2×C10, C2×C42, C2×C4⋊C4 [×6], C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C2×F5 [×12], C22×D5, C429C4, C4×Dic5 [×3], C4×C20, C4⋊F5 [×12], C2×C4×D5 [×3], C22×F5 [×4], D5×C42, C2×C4⋊F5 [×6], C428F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], F5, C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C2×F5 [×3], C429C4, C4⋊F5 [×6], C22×F5, C2×C4⋊F5 [×3], C428F5

Smallest permutation representation of C428F5
On 80 points
Generators in S80
```(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(21 31)(22 33 25 34)(23 35 24 32)(26 36)(27 38 30 39)(28 40 29 37)(41 46)(42 48 45 49)(43 50 44 47)(51 56)(52 58 55 59)(53 60 54 57)(61 76)(62 78 65 79)(63 80 64 77)(66 71)(67 73 70 74)(68 75 69 72)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 4 4 4 type + + + + - + + image C1 C2 C2 C4 C4 D4 Q8 F5 C2×F5 C4⋊F5 kernel C42⋊8F5 D5×C42 C2×C4⋊F5 C4×Dic5 C4×C20 C4×D5 C4×D5 C42 C2×C4 C4 # reps 1 1 6 6 2 6 6 1 3 12

Matrix representation of C428F5 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 5 0 0 0 0 0 0 9 35 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 5 0 0 0 0 0 0 16 40 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 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 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 9 0 0 0 0 0 0 0 21 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 3 32 0 0 0 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 0 0 0 0 40 40 40 40

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,9,0,0,0,0,0,0,5,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,16,0,0,0,0,0,0,5,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[9,21,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,3,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40] >;`

C428F5 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_8F_5`
`% in TeX`

`G:=Group("C4^2:8F5");`
`// GroupNames label`

`G:=SmallGroup(320,1026);`
`// by ID`

`G=gap.SmallGroup(320,1026);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,100,6278,1595]);`
`// Polycyclic`

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

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