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## G = C11⋊C5order 55 = 5·11

### The semidirect product of C11 and C5 acting faithfully

Aliases: C11⋊C5, SmallGroup(55,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C11⋊C5
 Chief series C1 — C11 — C11⋊C5
 Lower central C11 — C11⋊C5
 Upper central C1

Generators and relations for C11⋊C5
G = < a,b | a11=b5=1, bab-1=a3 >

Character table of C11⋊C5

 class 1 5A 5B 5C 5D 11A 11B size 1 11 11 11 11 5 5 ρ1 1 1 1 1 1 1 1 trivial ρ2 1 ζ5 ζ53 ζ52 ζ54 1 1 linear of order 5 ρ3 1 ζ54 ζ52 ζ53 ζ5 1 1 linear of order 5 ρ4 1 ζ52 ζ5 ζ54 ζ53 1 1 linear of order 5 ρ5 1 ζ53 ζ54 ζ5 ζ52 1 1 linear of order 5 ρ6 5 0 0 0 0 -1+√-11/2 -1-√-11/2 complex faithful ρ7 5 0 0 0 0 -1-√-11/2 -1+√-11/2 complex faithful

Permutation representations of C11⋊C5
On 11 points: primitive - transitive group 11T3
Generators in S11
```(1 2 3 4 5 6 7 8 9 10 11)
(2 5 6 10 4)(3 9 11 8 7)```

`G:=sub<Sym(11)| (1,2,3,4,5,6,7,8,9,10,11), (2,5,6,10,4)(3,9,11,8,7)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11), (2,5,6,10,4)(3,9,11,8,7) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11)], [(2,5,6,10,4),(3,9,11,8,7)]])`

`G:=TransitiveGroup(11,3);`

C11⋊C5 is a maximal subgroup of   F11
C11⋊C5 is a maximal quotient of   C11⋊C25

Polynomial with Galois group C11⋊C5 over ℚ
actionf(x)Disc(f)
11T3x11-33x9+396x7-2079x5+4455x3-2673x-243350·1116

Matrix representation of C11⋊C5 in GL5(𝔽3)

 2 2 2 0 0 2 2 1 1 1 2 1 0 0 2 0 1 1 0 1 2 1 2 0 1
,
 1 0 2 0 0 0 0 2 0 1 0 0 1 1 2 0 1 1 0 1 0 0 1 0 1

`G:=sub<GL(5,GF(3))| [2,2,2,0,2,2,2,1,1,1,2,1,0,1,2,0,1,0,0,0,0,1,2,1,1],[1,0,0,0,0,0,0,0,1,0,2,2,1,1,1,0,0,1,0,0,0,1,2,1,1] >;`

C11⋊C5 in GAP, Magma, Sage, TeX

`C_{11}\rtimes C_5`
`% in TeX`

`G:=Group("C11:C5");`
`// GroupNames label`

`G:=SmallGroup(55,1);`
`// by ID`

`G=gap.SmallGroup(55,1);`
`# by ID`

`G:=PCGroup([2,-5,-11,81]);`
`// Polycyclic`

`G:=Group<a,b|a^11=b^5=1,b*a*b^-1=a^3>;`
`// generators/relations`

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