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## G = C11⋊C40order 440 = 23·5·11

### The semidirect product of C11 and C40 acting via C40/C4=C10

Aliases: C11⋊C40, C22.C20, C4.2F11, C44.2C10, C11⋊C8⋊C5, C11⋊C5⋊C8, C2.(C11⋊C20), (C2×C11⋊C5).C4, (C4×C11⋊C5).2C2, SmallGroup(440,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C11⋊C40
 Chief series C1 — C11 — C22 — C44 — C4×C11⋊C5 — C11⋊C40
 Lower central C11 — C11⋊C40
 Upper central C1 — C4

Generators and relations for C11⋊C40
G = < a,b | a11=b40=1, bab-1=a8 >

Smallest permutation representation of C11⋊C40
On 88 points
Generators in S88
```(1 24 69 48 40 16 53 77 85 32 61)(2 86 17 70 62 78 41 25 33 54 9)(3 34 79 18 10 26 63 87 55 42 71)(4 56 27 80 72 88 11 35 43 64 19)(5 44 49 28 20 36 73 57 65 12 81)(6 66 37 50 82 58 21 45 13 74 29)(7 14 59 38 30 46 83 67 75 22 51)(8 76 47 60 52 68 31 15 23 84 39)
(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 81 82 83 84 85 86 87 88)```

`G:=sub<Sym(88)| (1,24,69,48,40,16,53,77,85,32,61)(2,86,17,70,62,78,41,25,33,54,9)(3,34,79,18,10,26,63,87,55,42,71)(4,56,27,80,72,88,11,35,43,64,19)(5,44,49,28,20,36,73,57,65,12,81)(6,66,37,50,82,58,21,45,13,74,29)(7,14,59,38,30,46,83,67,75,22,51)(8,76,47,60,52,68,31,15,23,84,39), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)>;`

`G:=Group( (1,24,69,48,40,16,53,77,85,32,61)(2,86,17,70,62,78,41,25,33,54,9)(3,34,79,18,10,26,63,87,55,42,71)(4,56,27,80,72,88,11,35,43,64,19)(5,44,49,28,20,36,73,57,65,12,81)(6,66,37,50,82,58,21,45,13,74,29)(7,14,59,38,30,46,83,67,75,22,51)(8,76,47,60,52,68,31,15,23,84,39), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88) );`

`G=PermutationGroup([[(1,24,69,48,40,16,53,77,85,32,61),(2,86,17,70,62,78,41,25,33,54,9),(3,34,79,18,10,26,63,87,55,42,71),(4,56,27,80,72,88,11,35,43,64,19),(5,44,49,28,20,36,73,57,65,12,81),(6,66,37,50,82,58,21,45,13,74,29),(7,14,59,38,30,46,83,67,75,22,51),(8,76,47,60,52,68,31,15,23,84,39)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)]])`

44 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 11 20A ··· 20H 22 40A ··· 40P 44A 44B order 1 2 4 4 5 5 5 5 8 8 8 8 10 10 10 10 11 20 ··· 20 22 40 ··· 40 44 44 size 1 1 1 1 11 11 11 11 11 11 11 11 11 11 11 11 10 11 ··· 11 10 11 ··· 11 10 10

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 10 10 10 type + + + - image C1 C2 C4 C5 C8 C10 C20 C40 F11 C11⋊C20 C11⋊C40 kernel C11⋊C40 C4×C11⋊C5 C2×C11⋊C5 C11⋊C8 C11⋊C5 C44 C22 C11 C4 C2 C1 # reps 1 1 2 4 4 4 8 16 1 1 2

Matrix representation of C11⋊C40 in GL10(𝔽881)

 880 1 0 0 0 0 0 0 0 0 880 0 1 0 0 0 0 0 0 0 880 0 0 1 0 0 0 0 0 0 880 0 0 0 1 0 0 0 0 0 880 0 0 0 0 1 0 0 0 0 880 0 0 0 0 0 1 0 0 0 880 0 0 0 0 0 0 1 0 0 880 0 0 0 0 0 0 0 1 0 880 0 0 0 0 0 0 0 0 1 880 0 0 0 0 0 0 0 0 0
,
 0 560 321 321 321 560 0 814 0 0 321 0 0 321 254 560 0 560 0 560 321 814 0 321 0 560 560 0 321 0 321 560 0 0 321 0 0 560 321 814 0 0 321 321 0 0 814 560 321 560 321 560 321 254 0 0 560 560 0 0 254 560 321 0 0 560 0 0 321 560 0 560 0 321 321 0 560 0 254 560 321 0 321 0 321 814 560 0 0 560 0 0 254 0 321 560 560 560 321 0

`G:=sub<GL(10,GF(881))| [880,880,880,880,880,880,880,880,880,880,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[0,321,321,321,0,321,254,0,321,0,560,0,814,560,0,560,560,560,0,0,321,0,0,0,321,321,321,0,321,254,321,321,321,0,321,254,0,321,0,0,321,254,0,321,0,0,0,321,321,321,560,560,560,0,0,0,560,0,814,560,0,0,560,0,814,560,0,560,560,560,814,560,0,560,560,560,0,0,0,560,0,0,321,321,321,0,321,254,0,321,0,560,0,814,560,0,560,560,560,0] >;`

C11⋊C40 in GAP, Magma, Sage, TeX

`C_{11}\rtimes C_{40}`
`% in TeX`

`G:=Group("C11:C40");`
`// GroupNames label`

`G:=SmallGroup(440,1);`
`// by ID`

`G=gap.SmallGroup(440,1);`
`# by ID`

`G:=PCGroup([5,-2,-5,-2,-2,-11,50,42,10004,4509]);`
`// Polycyclic`

`G:=Group<a,b|a^11=b^40=1,b*a*b^-1=a^8>;`
`// generators/relations`

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