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## G = C20.23C42order 320 = 26·5

### 16th non-split extension by C20 of C42 acting via C42/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.23C42
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — C20.23C42
 Lower central C5 — C20 — C20.23C42
 Upper central C1 — C4 — C2×C8

Generators and relations for C20.23C42
G = < a,b,c | a20=1, b4=a5, c4=a10, bab-1=a13, ac=ca, cbc-1=a5b >

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 8E 8F 10A 10B 10C 16A ··· 16H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 4 4 4 4 4 5 8 8 8 8 8 8 10 10 10 16 ··· 16 20 20 20 20 40 ··· 40 size 1 1 2 1 1 2 20 20 4 4 4 10 10 10 10 4 4 4 20 ··· 20 4 4 4 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 type + + + + + - image C1 C2 C2 C4 C4 C4 M4(2) M4(2) F5 C2×F5 C16⋊C4 C4×F5 C22.F5 C4.F5 C20.23C42 kernel C20.23C42 C40⋊8C4 C20.C8 C5⋊C16 C4×Dic5 C2×C40 C20 C2×C10 C2×C8 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 2 8 2 2 2 2 1 1 2 2 2 2 8

Matrix representation of C20.23C42 in GL8(𝔽241)

 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 240 0 0 1 0 0 0 0 0 240 0 1 0 0 0 0 0 0 240 1
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 141 20 141 90 0 0 0 0 41 110 231 231 0 0 0 0 131 10 10 131 0 0 0 0 151 151 100 221
,
 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 17 207 0 34 0 0 0 0 0 224 207 34 0 0 0 0 34 207 224 0 0 0 0 0 34 0 207 17

`G:=sub<GL(8,GF(241))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1],[0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,141,41,131,151,0,0,0,0,20,110,10,151,0,0,0,0,141,231,10,100,0,0,0,0,90,231,131,221],[0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,17,0,34,34,0,0,0,0,207,224,207,0,0,0,0,0,0,207,224,207,0,0,0,0,34,34,0,17] >;`

C20.23C42 in GAP, Magma, Sage, TeX

`C_{20}._{23}C_4^2`
`% in TeX`

`G:=Group("C20.23C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,228);`
`// by ID`

`G=gap.SmallGroup(320,228);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,100,1123,136,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=a^5,c^4=a^10,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^5*b>;`
`// generators/relations`

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