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G = C3⋊D4order 24 = 23·3

The semidirect product of C3 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C32D4, D62C2, Dic3⋊C2, C2.5D6, C222S3, C6.5C22, (C2×C6)⋊2C2, SmallGroup(24,8)

Series: Derived Chief Lower central Upper central

C1C6 — C3⋊D4
C1C3C6D6 — C3⋊D4
C3C6 — C3⋊D4
C1C2C22

Generators and relations for C3⋊D4
 G = < a,b,c | a3=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
6C2
3C4
3C22
2S3
2C6
3D4

Character table of C3⋊D4

 class 12A2B2C346A6B6C
 size 112626222
ρ1111111111    trivial
ρ2111-11-1111    linear of order 2
ρ311-111-1-1-11    linear of order 2
ρ411-1-111-1-11    linear of order 2
ρ522-20-1011-1    orthogonal lifted from D6
ρ62220-10-1-1-1    orthogonal lifted from S3
ρ72-2002000-2    orthogonal lifted from D4
ρ82-200-10--3-31    complex faithful
ρ92-200-10-3--31    complex faithful

Permutation representations of C3⋊D4
On 12 points - transitive group 12T13
Generators in S12
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)

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

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

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

G:=TransitiveGroup(12,13);

On 12 points - transitive group 12T15
Generators in S12
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)

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

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

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

G:=TransitiveGroup(12,15);

Regular action on 24 points - transitive group 24T14
Generators in S24
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,14);

Polynomial with Galois group C3⋊D4 over ℚ
actionf(x)Disc(f)
12T13x12-9x8+3x4-3-256·315
12T15x12-24x10+216x8-896x6+1680x4-1152x2+48268·317·136

Matrix representation of C3⋊D4 in GL2(𝔽7) generated by

40
02
,
06
10
,
01
10
G:=sub<GL(2,GF(7))| [4,0,0,2],[0,1,6,0],[0,1,1,0] >;

C3⋊D4 in GAP, Magma, Sage, TeX

C_3\rtimes D_4
% in TeX

G:=Group("C3:D4");
// GroupNames label

G:=SmallGroup(24,8);
// by ID

G=gap.SmallGroup(24,8);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,49,259]);
// Polycyclic

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

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