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G = C20.32C42order 320 = 26·5

2nd non-split extension by C20 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.32C42
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C2×C4.Dic5 — C20.32C42
 Lower central C5 — C10 — C20 — C20.32C42
 Upper central C1 — C2×C4 — C22×C4 — C42⋊C2

Generators and relations for C20.32C42
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, cac-1=a11, cbc-1=a5b >

Subgroups: 326 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C52C8, C2×Dic5, C2×C20, C2×C20, C22×C10, C426C4, C2×C52C8, C4.Dic5, C4.Dic5, C4×Dic5, C4×Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C2×C4.Dic5, C2×C4×Dic5, C5×C42⋊C2, C20.32C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4≀C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C426C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, D42Dic5, C20.32C42

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + - + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 D5 Dic5 D10 C4≀C2 Dic10 C4×D5 D20 C5⋊D4 C5⋊D4 D4⋊2Dic5 kernel C20.32C42 C2×C4.Dic5 C2×C4×Dic5 C5×C42⋊C2 C4.Dic5 C4×Dic5 C5×C4⋊C4 C2×C20 C2×C20 C22×C10 C42⋊C2 C4⋊C4 C22×C4 C10 C2×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 2 1 1 2 4 2 8 4 8 4 4 4 8

Matrix representation of C20.32C42 in GL4(𝔽41) generated by

 9 6 0 0 0 32 0 0 0 0 35 1 0 0 40 0
,
 15 38 0 0 37 26 0 0 0 0 39 28 0 0 16 2
,
 26 5 0 0 4 15 0 0 0 0 2 13 0 0 28 39
`G:=sub<GL(4,GF(41))| [9,0,0,0,6,32,0,0,0,0,35,40,0,0,1,0],[15,37,0,0,38,26,0,0,0,0,39,16,0,0,28,2],[26,4,0,0,5,15,0,0,0,0,2,28,0,0,13,39] >;`

C20.32C42 in GAP, Magma, Sage, TeX

`C_{20}._{32}C_4^2`
`% in TeX`

`G:=Group("C20.32C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,90);`
`// by ID`

`G=gap.SmallGroup(320,90);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1684,851,102,12550]);`
`// Polycyclic`

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

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