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## G = C20⋊(C4⋊C4)  order 320 = 26·5

### The semidirect product of C20 and C4⋊C4 acting via C4⋊C4/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20⋊(C4⋊C4)
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4×F5 — C20⋊(C4⋊C4)
 Lower central C5 — C2×C10 — C20⋊(C4⋊C4)
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C20⋊(C4⋊C4)
G = < a,b,c | a20=b4=c4=1, bab-1=a-1, cac-1=a13, cbc-1=b-1 >

Subgroups: 666 in 170 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×25], C23, D5 [×4], C10 [×3], C42 [×2], C4⋊C4, C4⋊C4 [×9], C22×C4 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×6], D10 [×6], C2×C10, C2.C42 [×2], C2×C42, C2×C4⋊C4 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×F5 [×4], C2×F5 [×10], C22×D5, C23.65C23, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C4×F5 [×2], C4⋊F5 [×6], C2×C4×D5, C2×C4×D5 [×2], C22×F5 [×2], C22×F5 [×2], D10.3Q8 [×2], D5×C4⋊C4, C2×C4×F5, C2×C4⋊F5, C2×C4⋊F5 [×2], C20⋊(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], F5, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×F5 [×3], C23.65C23, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D4×F5, Q8×F5, C20⋊(C4⋊C4)

Smallest permutation representation of C20⋊(C4⋊C4)
On 80 points
Generators in S80
```(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 36 59 68)(2 35 60 67)(3 34 41 66)(4 33 42 65)(5 32 43 64)(6 31 44 63)(7 30 45 62)(8 29 46 61)(9 28 47 80)(10 27 48 79)(11 26 49 78)(12 25 50 77)(13 24 51 76)(14 23 52 75)(15 22 53 74)(16 21 54 73)(17 40 55 72)(18 39 56 71)(19 38 57 70)(20 37 58 69)
(1 31)(2 28 10 24)(3 25 19 37)(4 22 8 30)(5 39 17 23)(6 36)(7 33 15 29)(9 27 13 35)(11 21)(12 38 20 34)(14 32 18 40)(16 26)(41 77 57 69)(42 74 46 62)(43 71 55 75)(44 68)(45 65 53 61)(47 79 51 67)(48 76 60 80)(49 73)(50 70 58 66)(52 64 56 72)(54 78)(59 63)```

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

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

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

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 2 2 4 4 4 8 8 type + + + + + + - + - + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 D4 Q8 C4○D4 F5 C2×F5 C4⋊F5 D4×F5 Q8×F5 kernel C20⋊(C4⋊C4) D10.3Q8 D5×C4⋊C4 C2×C4×F5 C2×C4⋊F5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C4×D5 C4×D5 C2×F5 C2×F5 D10 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 4 2 2 2 2 2 2 4 1 3 4 1 1

Matrix representation of C20⋊(C4⋊C4) in GL6(𝔽41)

 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 27 0 14 34 0 0 27 14 7 0 0 0 0 7 14 27 0 0 34 14 0 27
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 14 27 34 0 0 0 7 27 0 14 0 0 14 0 27 7 0 0 0 34 27 14

`G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,7,14,0,0,0,27,27,0,34,0,0,34,0,27,27,0,0,0,14,7,14] >;`

C20⋊(C4⋊C4) in GAP, Magma, Sage, TeX

`C_{20}\rtimes (C_4\rtimes C_4)`
`% in TeX`

`G:=Group("C20:(C4:C4)");`
`// GroupNames label`

`G:=SmallGroup(320,1050);`
`// by ID`

`G=gap.SmallGroup(320,1050);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,100,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=b^-1>;`
`// generators/relations`

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