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## G = C20.51C42order 320 = 26·5

### 14th non-split extension by C20 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.51C42
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C2×C4.Dic5 — C20.51C42
 Lower central C5 — C20 — C20.51C42
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C20.51C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a9, ac=ca, cbc-1=a5b >

Subgroups: 214 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×M4(2), C52C8, C40, C2×C20, C2×C20, C22×C10, C4.10C42, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C22×C20, C2×C4.Dic5, C10×M4(2), C20.51C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4.10C42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, C20.51C42

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E ··· 8L 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 2 2 1 1 2 2 2 2 2 4 4 4 4 20 ··· 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + - + + - image C1 C2 C2 C4 C4 D4 Q8 D5 Dic5 D10 C4×D5 D20 C5⋊D4 Dic10 C4.10C42 C20.51C42 kernel C20.51C42 C2×C4.Dic5 C10×M4(2) C2×C5⋊2C8 C2×C40 C2×C20 C22×C10 C2×M4(2) C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C5 C1 # reps 1 2 1 8 4 3 1 2 4 2 8 4 8 4 2 8

Matrix representation of C20.51C42 in GL6(𝔽41)

 40 40 0 0 0 0 8 7 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 26 3 0 0 0 0 21 15 0 0 0 0 0 0 27 0 9 0 0 0 11 0 20 1 0 0 0 0 14 0 0 0 32 9 31 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 21 9 0 0 0 0 32 20 0 0 0 0 29 14 0 9 0 0 7 31 1 0

`G:=sub<GL(6,GF(41))| [40,8,0,0,0,0,40,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[26,21,0,0,0,0,3,15,0,0,0,0,0,0,27,11,0,32,0,0,0,0,0,9,0,0,9,20,14,31,0,0,0,1,0,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,32,29,7,0,0,9,20,14,31,0,0,0,0,0,1,0,0,0,0,9,0] >;`

C20.51C42 in GAP, Magma, Sage, TeX

`C_{20}._{51}C_4^2`
`% in TeX`

`G:=Group("C20.51C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,118);`
`// by ID`

`G=gap.SmallGroup(320,118);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1123,136,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^9,a*c=c*a,c*b*c^-1=a^5*b>;`
`// generators/relations`

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