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G = C2×C7⋊C3order 42 = 2·3·7

Direct product of C2 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C7 — C2×C7⋊C3
C1C7C7⋊C3 — C2×C7⋊C3
C7 — C2×C7⋊C3
C1C2

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

7C3
7C6

Character table of C2×C7⋊C3

 class 123A3B6A6B7A7B14A14B
 size 1177773333
ρ11111111111    trivial
ρ21-111-1-111-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ321111    linear of order 3
ρ411ζ3ζ32ζ32ζ31111    linear of order 3
ρ51-1ζ3ζ32ζ6ζ6511-1-1    linear of order 6
ρ61-1ζ32ζ3ζ65ζ611-1-1    linear of order 6
ρ7330000-1+-7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ83-30000-1+-7/2-1--7/21+-7/21--7/2    complex faithful
ρ93-30000-1--7/2-1+-7/21--7/21+-7/2    complex faithful
ρ10330000-1--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);

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

1000
0100
0010
,
087
1033
083
,
188
038
087
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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