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G = C2×C11⋊C5order 110 = 2·5·11

Direct product of C2 and C11⋊C5

Aliases: C2×C11⋊C5, C22⋊C5, C112C10, SmallGroup(110,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C2×C11⋊C5
 Chief series C1 — C11 — C11⋊C5 — C2×C11⋊C5
 Lower central C11 — C2×C11⋊C5
 Upper central C1 — C2

Generators and relations for C2×C11⋊C5
G = < a,b,c | a2=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

Character table of C2×C11⋊C5

 class 1 2 5A 5B 5C 5D 10A 10B 10C 10D 11A 11B 22A 22B size 1 1 11 11 11 11 11 11 11 11 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 -1 ζ52 ζ53 ζ5 ζ54 -ζ52 -ζ53 -ζ5 -ζ54 1 1 -1 -1 linear of order 10 ρ4 1 1 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ54 ζ5 1 1 1 1 linear of order 5 ρ5 1 -1 ζ54 ζ5 ζ52 ζ53 -ζ54 -ζ5 -ζ52 -ζ53 1 1 -1 -1 linear of order 10 ρ6 1 1 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ53 ζ52 1 1 1 1 linear of order 5 ρ7 1 -1 ζ5 ζ54 ζ53 ζ52 -ζ5 -ζ54 -ζ53 -ζ52 1 1 -1 -1 linear of order 10 ρ8 1 1 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ5 ζ54 1 1 1 1 linear of order 5 ρ9 1 1 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ52 ζ53 1 1 1 1 linear of order 5 ρ10 1 -1 ζ53 ζ52 ζ54 ζ5 -ζ53 -ζ52 -ζ54 -ζ5 1 1 -1 -1 linear of order 10 ρ11 5 5 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 -1-√-11/2 -1+√-11/2 complex lifted from C11⋊C5 ρ12 5 -5 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 1-√-11/2 1+√-11/2 complex faithful ρ13 5 -5 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 1+√-11/2 1-√-11/2 complex faithful ρ14 5 5 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 -1+√-11/2 -1-√-11/2 complex lifted from C11⋊C5

Permutation representations of C2×C11⋊C5
On 22 points - transitive group 22T5
Generators in S22
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)

G:=sub<Sym(22)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)>;

G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18) );

G=PermutationGroup([(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22)], [(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18)])

G:=TransitiveGroup(22,5);

C2×C11⋊C5 is a maximal subgroup of   C11⋊C20

Matrix representation of C2×C11⋊C5 in GL5(𝔽3)

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

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

C2×C11⋊C5 in GAP, Magma, Sage, TeX

C_2\times C_{11}\rtimes C_5
% in TeX

G:=Group("C2xC11:C5");
// GroupNames label

G:=SmallGroup(110,2);
// by ID

G=gap.SmallGroup(110,2);
# by ID

G:=PCGroup([3,-2,-5,-11,185]);
// Polycyclic

G:=Group<a,b,c|a^2=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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