metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊2D4, D6⋊2C2, Dic3⋊C2, C2.5D6, C22⋊2S3, C6.5C22, (C2×C6)⋊2C2, SmallGroup(24,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D4
G = < a,b,c | a3=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C3⋊D4
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | |
| size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | -1 | 0 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | -2 | 0 | 0 | -1 | 0 | -√-3 | √-3 | 1 | complex faithful |
| ρ9 | 2 | -2 | 0 | 0 | -1 | 0 | √-3 | -√-3 | 1 | complex faithful |
►On 12 points - transitive group 12T13
(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
(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
(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);
C3⋊D4 is a maximal subgroup of
C3.S4 C32⋊7D4 C3⋊S4 Q8.D6 A4⋊D4 C33⋊D4 C2.S5
D6p⋊C2: C4○D12 S3×D4 C9⋊D4 C3⋊D20 C15⋊7D4 C3⋊D28 C21⋊7D4 C3⋊D44 ...
D2p⋊S3: D4⋊2S3 D6⋊S3 C3⋊D12 C15⋊D4 C21⋊D4 C33⋊D4 C39⋊D4 C51⋊D4 ...
C3⋊D4 is a maximal quotient of
A4⋊D4 C33⋊D4
C3⋊D4p: D4⋊S3 C3⋊D12 C3⋊D20 C3⋊D28 C3⋊D44 C3⋊D52 C3⋊D68 C3⋊D76 ...
C6.D2p: Dic3⋊C4 D6⋊C4 D4.S3 Q8⋊2S3 C3⋊Q16 C6.D4 C9⋊D4 D6⋊S3 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 12T13 | x12-9x8+3x4-3 | -256·315 |
| 12T15 | x12-24x10+216x8-896x6+1680x4-1152x2+48 | 268·317·136 |
Matrix representation of C3⋊D4 ►in GL2(𝔽7) generated by
| 4 | 0 |
| 0 | 2 |
| 0 | 6 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
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
Export
Subgroup lattice of C3⋊D4 in TeX
Character table of C3⋊D4 in TeX