Copied to
clipboard

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

C1C2 — C3×D4
C1C2C6C2×C6 — C3×D4
C1C2 — C3×D4
C1C6 — C3×D4

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

2C2
2C2
2C6
2C6

Character table of C3×D4

 class 12A2B2C3A3B46A6B6C6D6E6F12A12B
 size 112211211222222
ρ1111111111111111    trivial
ρ211-1-111111-1-1-1-111    linear of order 2
ρ311-1111-111-111-1-1-1    linear of order 2
ρ4111-111-1111-1-11-1-1    linear of order 2
ρ511-1-1ζ3ζ321ζ32ζ3ζ6ζ6ζ65ζ65ζ32ζ3    linear of order 6
ρ6111-1ζ32ζ3-1ζ3ζ32ζ3ζ65ζ6ζ32ζ65ζ6    linear of order 6
ρ711-1-1ζ32ζ31ζ3ζ32ζ65ζ65ζ6ζ6ζ3ζ32    linear of order 6
ρ811-11ζ3ζ32-1ζ32ζ3ζ6ζ32ζ3ζ65ζ6ζ65    linear of order 6
ρ91111ζ32ζ31ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ1011-11ζ32ζ3-1ζ3ζ32ζ65ζ3ζ32ζ6ζ65ζ6    linear of order 6
ρ11111-1ζ3ζ32-1ζ32ζ3ζ32ζ6ζ65ζ3ζ6ζ65    linear of order 6
ρ121111ζ3ζ321ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ132-200220-2-2000000    orthogonal lifted from D4
ρ142-200-1--3-1+-301--31+-3000000    complex faithful
ρ152-200-1+-3-1--301+-31--3000000    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

40
04
,
06
10
,
01
10
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

׿
×
𝔽