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## G = C3×D4order 24 = 23·3

### Direct product of C3 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×D4, C4⋊C6, C123C2, C222C6, C6.6C22, (C2×C6)⋊1C2, C2.1(C2×C6), SmallGroup(24,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×D4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4
 Lower central C1 — C2 — C3×D4
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4
G = < a,b,c | a3=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C3×D4

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 12A 12B size 1 1 2 2 1 1 2 1 1 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 linear of order 6 ρ6 1 1 1 -1 ζ32 ζ3 -1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 linear of order 6 ρ7 1 1 -1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 linear of order 6 ρ8 1 1 -1 1 ζ3 ζ32 -1 ζ32 ζ3 ζ6 ζ32 ζ3 ζ65 ζ6 ζ65 linear of order 6 ρ9 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ10 1 1 -1 1 ζ32 ζ3 -1 ζ3 ζ32 ζ65 ζ3 ζ32 ζ6 ζ65 ζ6 linear of order 6 ρ11 1 1 1 -1 ζ3 ζ32 -1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 linear of order 6 ρ12 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ13 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 0 0 -1-√-3 -1+√-3 0 1-√-3 1+√-3 0 0 0 0 0 0 complex faithful ρ15 2 -2 0 0 -1+√-3 -1-√-3 0 1+√-3 1-√-3 0 0 0 0 0 0 complex faithful

Permutation representations of C3×D4
On 12 points - transitive group 12T14
Generators in S12
(1 7 9)(2 8 10)(3 5 11)(4 6 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)

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

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

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

G:=TransitiveGroup(12,14);

Regular action on 24 points - transitive group 24T15
Generators in S24
(1 19 13)(2 20 14)(3 17 15)(4 18 16)(5 9 23)(6 10 24)(7 11 21)(8 12 22)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 24)(18 23)(19 22)(20 21)

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

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

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

G:=TransitiveGroup(24,15);

Polynomial with Galois group C3×D4 over ℚ
actionf(x)Disc(f)
12T14x12-24x10+226x8-1056x6+2552x4-3008x2+1352257·710·132

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

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

C3×D4 in GAP, Magma, Sage, TeX

C_3\times D_4
% in TeX

G:=Group("C3xD4");
// GroupNames label

G:=SmallGroup(24,10);
// by ID

G=gap.SmallGroup(24,10);
# by ID

G:=PCGroup([4,-2,-2,-3,-2,113]);
// Polycyclic

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

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