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## G = C2×F11order 220 = 22·5·11

### Direct product of C2 and F11

Aliases: C2×F11, D22⋊C5, C22⋊C10, D11⋊C10, C11⋊(C2×C10), C11⋊C5⋊C22, (C2×C11⋊C5)⋊C2, Aut(D22), Hol(C22), SmallGroup(220,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C2×F11
 Chief series C1 — C11 — C11⋊C5 — F11 — C2×F11
 Lower central C11 — C2×F11
 Upper central C1 — C2

Generators and relations for C2×F11
G = < a,b,c | a2=b11=c10=1, ab=ba, ac=ca, cbc-1=b6 >

Character table of C2×F11

 class 1 2A 2B 2C 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 11 22 size 1 1 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 10 10 ρ1 1 1 1 1 1 1 1 1 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 -1 -1 1 1 1 -1 1 -1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 linear of order 2 ρ5 1 -1 1 -1 ζ5 ζ54 ζ53 ζ52 ζ5 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 -ζ53 ζ54 ζ53 ζ52 -ζ52 1 -1 linear of order 10 ρ6 1 -1 1 -1 ζ54 ζ5 ζ52 ζ53 ζ54 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 -ζ52 ζ5 ζ52 ζ53 -ζ53 1 -1 linear of order 10 ρ7 1 -1 -1 1 ζ53 ζ52 ζ54 ζ5 -ζ53 -ζ53 -ζ52 ζ52 ζ54 ζ5 ζ53 -ζ54 -ζ52 -ζ54 -ζ5 -ζ5 1 -1 linear of order 10 ρ8 1 -1 1 -1 ζ53 ζ52 ζ54 ζ5 ζ53 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 -ζ54 ζ52 ζ54 ζ5 -ζ5 1 -1 linear of order 10 ρ9 1 -1 -1 1 ζ54 ζ5 ζ52 ζ53 -ζ54 -ζ54 -ζ5 ζ5 ζ52 ζ53 ζ54 -ζ52 -ζ5 -ζ52 -ζ53 -ζ53 1 -1 linear of order 10 ρ10 1 1 -1 -1 ζ5 ζ54 ζ53 ζ52 -ζ5 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 ζ53 -ζ54 -ζ53 -ζ52 ζ52 1 1 linear of order 10 ρ11 1 1 1 1 ζ5 ζ54 ζ53 ζ52 ζ5 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 ζ53 ζ54 ζ53 ζ52 ζ52 1 1 linear of order 5 ρ12 1 1 -1 -1 ζ52 ζ53 ζ5 ζ54 -ζ52 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 ζ5 -ζ53 -ζ5 -ζ54 ζ54 1 1 linear of order 10 ρ13 1 1 -1 -1 ζ53 ζ52 ζ54 ζ5 -ζ53 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 ζ54 -ζ52 -ζ54 -ζ5 ζ5 1 1 linear of order 10 ρ14 1 1 1 1 ζ54 ζ5 ζ52 ζ53 ζ54 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 ζ52 ζ5 ζ52 ζ53 ζ53 1 1 linear of order 5 ρ15 1 1 1 1 ζ52 ζ53 ζ5 ζ54 ζ52 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 ζ5 ζ53 ζ5 ζ54 ζ54 1 1 linear of order 5 ρ16 1 1 1 1 ζ53 ζ52 ζ54 ζ5 ζ53 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 ζ54 ζ52 ζ54 ζ5 ζ5 1 1 linear of order 5 ρ17 1 -1 -1 1 ζ52 ζ53 ζ5 ζ54 -ζ52 -ζ52 -ζ53 ζ53 ζ5 ζ54 ζ52 -ζ5 -ζ53 -ζ5 -ζ54 -ζ54 1 -1 linear of order 10 ρ18 1 1 -1 -1 ζ54 ζ5 ζ52 ζ53 -ζ54 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 ζ52 -ζ5 -ζ52 -ζ53 ζ53 1 1 linear of order 10 ρ19 1 -1 -1 1 ζ5 ζ54 ζ53 ζ52 -ζ5 -ζ5 -ζ54 ζ54 ζ53 ζ52 ζ5 -ζ53 -ζ54 -ζ53 -ζ52 -ζ52 1 -1 linear of order 10 ρ20 1 -1 1 -1 ζ52 ζ53 ζ5 ζ54 ζ52 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 -ζ5 ζ53 ζ5 ζ54 -ζ54 1 -1 linear of order 10 ρ21 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from F11 ρ22 10 -10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 orthogonal faithful

Permutation representations of C2×F11
On 22 points - transitive group 22T6
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)
(1 12)(2 14 5 20 6 22 10 19 4 18)(3 16 9 17 11 21 8 15 7 13)

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

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

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

G:=TransitiveGroup(22,6);

C2×F11 is a maximal subgroup of   D44⋊C5  C22⋊F11
C2×F11 is a maximal quotient of   C4.F11  D44⋊C5  C22⋊F11

Matrix representation of C2×F11 in GL10(ℤ)

 -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 0 0 0 0 0 0 0 0 -1
,
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 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
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

G:=sub<GL(10,Integers())| [-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,-1,0,0,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1] >;

C2×F11 in GAP, Magma, Sage, TeX

C_2\times F_{11}
% in TeX

G:=Group("C2xF11");
// GroupNames label

G:=SmallGroup(220,7);
// by ID

G=gap.SmallGroup(220,7);
# by ID

G:=PCGroup([4,-2,-2,-5,-11,3203,731]);
// Polycyclic

G:=Group<a,b,c|a^2=b^11=c^10=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

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