Copied to
clipboard

G = C3×D4  order 24 = 23·3

Direct product of C3 and D4

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

Aliases: C3×D4, C4⋊C6, C12⋊3C2, C22⋊2C6, 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 >

C1
C2
2C2
2C2
C3
C4
C22
C22
C6
2C6
2C6
D4
C2×C6
C12
C2×C6
C3×D4

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 15)(2 20 16)(3 17 13)(4 18 14)(5 10 21)(6 11 22)(7 12 23)(8 9 24)

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

C3×D4 is a maximal subgroup of
 D4⋊S3  D4.S3  D4⋊2S3  D4.A4  C4⋊F7  Dic7⋊C6  D52⋊C3  D26⋊C6  D76⋊C3  D38⋊C6
C3×D4 is a maximal quotient of
 C4⋊F7  Dic7⋊C6  D52⋊C3  D26⋊C6  D76⋊C3  D38⋊C6

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
 

Export

Subgroup lattice of C3×D4 in TeX
Character table of C3×D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁