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G = C7⋊C3order 21 = 3·7

The semidirect product of C7 and C3 acting faithfully

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

Aliases: C7⋊C3, SmallGroup(21,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C3
C1C7 — C7⋊C3
C7 — C7⋊C3
C1

Generators and relations for C7⋊C3
 G = < a,b | a7=b3=1, bab-1=a4 >

7C3

Character table of C7⋊C3

 class 13A3B7A7B
 size 17733
ρ111111    trivial
ρ21ζ32ζ311    linear of order 3
ρ31ζ3ζ3211    linear of order 3
ρ4300-1--7/2-1+-7/2    complex faithful
ρ5300-1+-7/2-1--7/2    complex faithful

Permutation representations of C7⋊C3
On 7 points: primitive - transitive group 7T3
Generators in S7
(1 2 3 4 5 6 7)
(2 3 5)(4 7 6)

G:=sub<Sym(7)| (1,2,3,4,5,6,7), (2,3,5)(4,7,6)>;

G:=Group( (1,2,3,4,5,6,7), (2,3,5)(4,7,6) );

G=PermutationGroup([(1,2,3,4,5,6,7)], [(2,3,5),(4,7,6)])

G:=TransitiveGroup(7,3);

Regular action on 21 points - transitive group 21T2
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 19 10)(2 21 14)(3 16 11)(4 18 8)(5 20 12)(6 15 9)(7 17 13)

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

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

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

G:=TransitiveGroup(21,2);

Polynomial with Galois group C7⋊C3 over ℚ
actionf(x)Disc(f)
7T3x7-8x5-2x4+16x3+6x2-6x-226·734

Matrix representation of C7⋊C3 in GL3(𝔽2) generated by

110
001
100
,
100
001
011
G:=sub<GL(3,GF(2))| [1,0,1,1,0,0,0,1,0],[1,0,0,0,0,1,0,1,1] >;

C7⋊C3 in GAP, Magma, Sage, TeX

C_7\rtimes C_3
% in TeX

G:=Group("C7:C3");
// GroupNames label

G:=SmallGroup(21,1);
// by ID

G=gap.SmallGroup(21,1);
# by ID

G:=PCGroup([2,-3,-7,25]);
// Polycyclic

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

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