direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.D4, C23.C12, C12.58D4, M4(2)⋊3C6, C4.9(C3×D4), (C2×D4).2C6, (C6×D4).8C2, (C22×C6).1C4, (C3×M4(2))⋊9C2, C22.3(C2×C12), C6.22(C22⋊C4), (C2×C12).59C22, (C2×C4).1(C2×C6), (C2×C6).20(C2×C4), C2.4(C3×C22⋊C4), SmallGroup(96,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.D4
G = < a,b,c,d | a3=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Permutation representations of C3×C4.D4
►On 24 points - transitive group 24T90
(1 10 21)(2 11 22)(3 12 23)(4 13 24)(5 14 17)(6 15 18)(7 16 19)(8 9 20)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)
(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 6 3 4 5 2 7 8)(9 10 15 12 13 14 11 16)(17 22 19 20 21 18 23 24)
G:=sub<Sym(24)| (1,10,21)(2,11,22)(3,12,23)(4,13,24)(5,14,17)(6,15,18)(7,16,19)(8,9,20), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,6,3,4,5,2,7,8)(9,10,15,12,13,14,11,16)(17,22,19,20,21,18,23,24)>;
G:=Group( (1,10,21)(2,11,22)(3,12,23)(4,13,24)(5,14,17)(6,15,18)(7,16,19)(8,9,20), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,6,3,4,5,2,7,8)(9,10,15,12,13,14,11,16)(17,22,19,20,21,18,23,24) );
G=PermutationGroup([[(1,10,21),(2,11,22),(3,12,23),(4,13,24),(5,14,17),(6,15,18),(7,16,19),(8,9,20)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,6,3,4,5,2,7,8),(9,10,15,12,13,14,11,16),(17,22,19,20,21,18,23,24)]])
G:=TransitiveGroup(24,90);
C3×C4.D4 is a maximal subgroup of
C23.3D12 C23.4D12 M4(2).19D6 M4(2)⋊D6 D12⋊1D4 D12.2D4 D12.3D4
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | C3×D4 | C4.D4 | C3×C4.D4 |
| kernel | C3×C4.D4 | C3×M4(2) | C6×D4 | C4.D4 | C22×C6 | M4(2) | C2×D4 | C23 | C12 | C4 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 1 | 2 |
Matrix representation of C3×C4.D4 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 5 | 4 | 6 |
| 6 | 0 | 6 | 0 |
| 3 | 3 | 3 | 1 |
| 3 | 4 | 2 | 0 |
| 0 | 2 | 5 | 2 |
| 0 | 0 | 5 | 6 |
| 5 | 2 | 1 | 3 |
| 4 | 4 | 3 | 6 |
| 4 | 1 | 0 | 4 |
| 4 | 3 | 3 | 0 |
| 2 | 5 | 6 | 4 |
| 3 | 3 | 2 | 1 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,6,3,3,5,0,3,4,4,6,3,2,6,0,1,0],[0,0,5,4,2,0,2,4,5,5,1,3,2,6,3,6],[4,4,2,3,1,3,5,3,0,3,6,2,4,0,4,1] >;
C3×C4.D4 in GAP, Magma, Sage, TeX
C_3\times C_4.D_4
% in TeX
G:=Group("C3xC4.D4"); // GroupNames label
G:=SmallGroup(96,50);
// by ID
G=gap.SmallGroup(96,50);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,1090,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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