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## G = C20.C42order 320 = 26·5

### 2nd non-split extension by C20 of C42 acting via C42/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.C42
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C4×D5 — C2×C4.F5 — C20.C42
 Lower central C5 — C10 — C20 — C20.C42
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C20.C42
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a3, cac-1=a11, cbc-1=a5b >

Subgroups: 450 in 110 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C426C4, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C4.F5, C4.F5, C4×F5, C4×F5, C2×C5⋊C8, C2×C4×D5, C22×F5, C4⋊C47D5, C2×C4.F5, C2×C4×F5, C20.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C4≀C2, C2×F5, C426C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, D4⋊F5, Q82F5, C20.C42

Smallest permutation representation of C20.C42
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 51 31 61 11 41 21 71)(2 58 40 64 12 48 30 74)(3 45 29 67 13 55 39 77)(4 52 38 70 14 42 28 80)(5 59 27 73 15 49 37 63)(6 46 36 76 16 56 26 66)(7 53 25 79 17 43 35 69)(8 60 34 62 18 50 24 72)(9 47 23 65 19 57 33 75)(10 54 32 68 20 44 22 78)
(1 66 26 51)(2 77 27 42)(3 68 28 53)(4 79 29 44)(5 70 30 55)(6 61 31 46)(7 72 32 57)(8 63 33 48)(9 74 34 59)(10 65 35 50)(11 76 36 41)(12 67 37 52)(13 78 38 43)(14 69 39 54)(15 80 40 45)(16 71 21 56)(17 62 22 47)(18 73 23 58)(19 64 24 49)(20 75 25 60)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 5 8A 8B 8C 8D 10A 10B 10C 20A ··· 20F order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 5 8 8 8 8 10 10 10 20 ··· 20 size 1 1 1 1 10 10 2 2 4 4 5 5 5 5 10 ··· 10 20 20 4 20 20 20 20 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 4 4 8 8 type + + + + + - + + + + + - + image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 D4 D4 C4≀C2 F5 C2×F5 C4×F5 C4⋊F5 C22⋊F5 D4⋊F5 Q8⋊2F5 kernel C20.C42 C4⋊C4⋊7D5 C2×C4.F5 C2×C4×F5 C4⋊Dic5 C5×C4⋊C4 C4.F5 C4×F5 C4×D5 C4×D5 C2×Dic5 C22×D5 C10 C4⋊C4 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 4 4 1 1 1 1 8 1 1 2 2 2 1 1

Matrix representation of C20.C42 in GL8(𝔽41)

 32 0 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 0 1 0 0 0 0 40 0 0 1 0 0 0 0 0 40 0 1 0 0 0 0 0 0 40 1
,
 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 5 0 0 0 0 0 0 16 40 0 0 0 0 0 0 0 0 3 38 19 0 0 0 0 0 22 38 0 3 0 0 0 0 3 0 38 22 0 0 0 0 0 19 38 3
,
 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 9 4 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 19 3 0 38 0 0 0 0 0 22 3 38 0 0 0 0 38 3 22 0 0 0 0 0 38 0 3 19

`G:=sub<GL(8,GF(41))| [32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1],[0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,5,40,0,0,0,0,0,0,0,0,3,22,3,0,0,0,0,0,38,38,0,19,0,0,0,0,19,0,38,38,0,0,0,0,0,3,22,3],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,4,32,0,0,0,0,0,0,0,0,19,0,38,38,0,0,0,0,3,22,3,0,0,0,0,0,0,3,22,3,0,0,0,0,38,38,0,19] >;`

C20.C42 in GAP, Magma, Sage, TeX

`C_{20}.C_4^2`
`% in TeX`

`G:=Group("C20.C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,213);`
`// by ID`

`G=gap.SmallGroup(320,213);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1684,851,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^3,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;`
`// generators/relations`

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