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G = C3⋊C8order 24 = 23·3

The semidirect product of C3 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3⋊C8, C6.C4, C4.2S3, C2.Dic3, C12.2C2, SmallGroup(24,1)

Series: Derived Chief Lower central Upper central

C1C3 — C3⋊C8
C1C3C6C12 — C3⋊C8
C3 — C3⋊C8
C1C4

Generators and relations for C3⋊C8
 G = < a,b | a3=b8=1, bab-1=a-1 >

3C8

Character table of C3⋊C8

 class 1234A4B68A8B8C8D12A12B
 size 112112333322
ρ1111111111111    trivial
ρ2111111-1-1-1-111    linear of order 2
ρ3111-1-11i-i-ii-1-1    linear of order 4
ρ4111-1-11-iii-i-1-1    linear of order 4
ρ51-11i-i-1ζ85ζ83ζ87ζ8i-i    linear of order 8
ρ61-11i-i-1ζ8ζ87ζ83ζ85i-i    linear of order 8
ρ71-11-ii-1ζ83ζ85ζ8ζ87-ii    linear of order 8
ρ81-11-ii-1ζ87ζ8ζ85ζ83-ii    linear of order 8
ρ922-122-10000-1-1    orthogonal lifted from S3
ρ1022-1-2-2-1000011    symplectic lifted from Dic3, Schur index 2
ρ112-2-1-2i2i10000i-i    complex faithful
ρ122-2-12i-2i10000-ii    complex faithful

Permutation representations of C3⋊C8
Regular action on 24 points - transitive group 24T8
Generators in S24
(1 21 13)(2 14 22)(3 23 15)(4 16 24)(5 17 9)(6 10 18)(7 19 11)(8 12 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

G:=sub<Sym(24)| (1,21,13)(2,14,22)(3,23,15)(4,16,24)(5,17,9)(6,10,18)(7,19,11)(8,12,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;

G:=Group( (1,21,13)(2,14,22)(3,23,15)(4,16,24)(5,17,9)(6,10,18)(7,19,11)(8,12,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );

G=PermutationGroup([(1,21,13),(2,14,22),(3,23,15),(4,16,24),(5,17,9),(6,10,18),(7,19,11),(8,12,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)])

G:=TransitiveGroup(24,8);

Matrix representation of C3⋊C8 in GL2(𝔽5) generated by

41
40
,
43
21
G:=sub<GL(2,GF(5))| [4,4,1,0],[4,2,3,1] >;

C3⋊C8 in GAP, Magma, Sage, TeX

C_3\rtimes C_8
% in TeX

G:=Group("C3:C8");
// GroupNames label

G:=SmallGroup(24,1);
// by ID

G=gap.SmallGroup(24,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,8,21,259]);
// Polycyclic

G:=Group<a,b|a^3=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

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