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G = C11⋊C5order 55 = 5·11

The semidirect product of C11 and C5 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C11⋊C5, SmallGroup(55,1)

Series: Derived Chief Lower central Upper central

C1C11 — C11⋊C5
C1C11 — C11⋊C5
C11 — C11⋊C5
C1

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

11C5

Character table of C11⋊C5

 class 15A5B5C5D11A11B
 size 11111111155
ρ11111111    trivial
ρ21ζ5ζ53ζ52ζ5411    linear of order 5
ρ31ζ54ζ52ζ53ζ511    linear of order 5
ρ41ζ52ζ5ζ54ζ5311    linear of order 5
ρ51ζ53ζ54ζ5ζ5211    linear of order 5
ρ650000-1+-11/2-1--11/2    complex faithful
ρ750000-1--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);

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)

22200
22111
21002
01101
21201
,
10200
00201
00112
01101
00101

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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