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## G = C2×C7⋊C3order 42 = 2·3·7

### Direct product of C2 and C7⋊C3

Aliases: C2×C7⋊C3, C14⋊C3, C72C6, SmallGroup(42,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C7⋊C3
 Chief series C1 — C7 — C7⋊C3 — C2×C7⋊C3
 Lower central C7 — C2×C7⋊C3
 Upper central C1 — C2

Generators and relations for C2×C7⋊C3
G = < a,b,c | a2=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Character table of C2×C7⋊C3

 class 1 2 3A 3B 6A 6B 7A 7B 14A 14B size 1 1 7 7 7 7 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 linear of order 3 ρ4 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 linear of order 3 ρ5 1 -1 ζ3 ζ32 ζ6 ζ65 1 1 -1 -1 linear of order 6 ρ6 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 -1 -1 linear of order 6 ρ7 3 3 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ8 3 -3 0 0 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 1-√-7/2 complex faithful ρ9 3 -3 0 0 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 1+√-7/2 complex faithful ρ10 3 3 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3

Permutation representations of C2×C7⋊C3
On 14 points - transitive group 14T5
Generators in S14
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)

G:=sub<Sym(14)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (2,3,5)(4,7,6)(9,10,12)(11,14,13)>;

G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (2,3,5)(4,7,6)(9,10,12)(11,14,13) );

G=PermutationGroup([(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13)])

G:=TransitiveGroup(14,5);

C2×C7⋊C3 is a maximal subgroup of   C7⋊C12  C14.A4  C73F7  C74F7  SL2(𝔽7)
C2×C7⋊C3 is a maximal quotient of   C73F7  C74F7

Polynomial with Galois group C2×C7⋊C3 over ℚ
actionf(x)Disc(f)
14T5x14+28x12+308x10+1680x8+4704x6+6272x4+3136x2+484-240·720·1114

Matrix representation of C2×C7⋊C3 in GL3(𝔽11) generated by

 10 0 0 0 10 0 0 0 10
,
 0 8 7 10 3 3 0 8 3
,
 1 8 8 0 3 8 0 8 7
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[0,10,0,8,3,8,7,3,3],[1,0,0,8,3,8,8,8,7] >;

C2×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes C_3
% in TeX

G:=Group("C2xC7:C3");
// GroupNames label

G:=SmallGroup(42,2);
// by ID

G=gap.SmallGroup(42,2);
# by ID

G:=PCGroup([3,-2,-3,-7,59]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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