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## G = C20.45C42order 320 = 26·5

### 8th non-split extension by C20 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.45C42
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C20.4C8 — C20.45C42
 Lower central C5 — C20 — C20.45C42
 Upper central C1 — C4 — C8⋊C4

Generators and relations for C20.45C42
G = < a,b,c | a20=c4=1, b4=a5, bab-1=a9, ac=ca, cbc-1=a5b >

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - - + image C1 C2 C2 C4 C4 C4 D5 M4(2) M4(2) Dic5 Dic5 D10 C4×D5 C4.Dic5 C4.Dic5 C16⋊C4 C20.45C42 kernel C20.45C42 C20.4C8 C5×C8⋊C4 C5⋊2C16 C4×C20 C2×C40 C8⋊C4 C20 C2×C10 C42 C2×C8 C2×C8 C8 C4 C22 C5 C1 # reps 1 2 1 8 2 2 2 2 2 2 2 2 8 8 8 2 8

Matrix representation of C20.45C42 in GL6(𝔽241)

 98 0 0 0 0 0 37 91 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64
,
 229 113 0 0 0 0 180 12 0 0 0 0 0 0 226 37 13 176 0 0 0 0 0 1 0 0 36 156 1 208 0 0 160 67 58 14
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 40 179 130 0 0 0 240 0 0 0 0 0 0 64 164 0 0 0 0 0 177

`G:=sub<GL(6,GF(241))| [98,37,0,0,0,0,0,91,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[229,180,0,0,0,0,113,12,0,0,0,0,0,0,226,0,36,160,0,0,37,0,156,67,0,0,13,0,1,58,0,0,176,1,208,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,40,240,0,0,0,0,179,0,64,0,0,0,130,0,164,177] >;`

C20.45C42 in GAP, Magma, Sage, TeX

`C_{20}._{45}C_4^2`
`% in TeX`

`G:=Group("C20.45C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,24);`
`// by ID`

`G=gap.SmallGroup(320,24);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,100,1123,102,12550]);`
`// Polycyclic`

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

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