direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4≀C2, D4⋊2C12, C42⋊6C6, Q8⋊3C12, C12.66D4, M4(2)⋊4C6, (C3×D4)⋊5C4, (C3×Q8)⋊5C4, (C4×C12)⋊10C2, C4.3(C2×C12), C4○D4.3C6, (C2×C6).22D4, C4.17(C3×D4), C12.30(C2×C4), C22.3(C3×D4), C6.26(C22⋊C4), (C3×M4(2))⋊10C2, (C2×C12).116C22, (C2×C4).19(C2×C6), (C3×C4○D4).4C2, C2.8(C3×C22⋊C4), SmallGroup(96,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4≀C2
G = < a,b,c,d | a3=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Permutation representations of C3×C4≀C2
►On 24 points - transitive group 24T113
(1 11 7)(2 12 8)(3 9 5)(4 10 6)(13 21 17)(14 22 18)(15 23 19)(16 24 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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 20)(8 19)(9 22)(10 21)(11 24)(12 23)
(13 16 15 14)(17 20 19 18)(21 24 23 22)
G:=sub<Sym(24)| (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,22)(10,21)(11,24)(12,23), (13,16,15,14)(17,20,19,18)(21,24,23,22)>;
G:=Group( (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,22)(10,21)(11,24)(12,23), (13,16,15,14)(17,20,19,18)(21,24,23,22) );
G=PermutationGroup([[(1,11,7),(2,12,8),(3,9,5),(4,10,6),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,20),(8,19),(9,22),(10,21),(11,24),(12,23)], [(13,16,15,14),(17,20,19,18),(21,24,23,22)]])
G:=TransitiveGroup(24,113);
C3×C4≀C2 is a maximal subgroup of
C42⋊3D6 Q8⋊5D12 M4(2).22D6 C42.196D6 C42⋊5D6 Q8.14D12 D4.10D12
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | C3×D4 | C3×D4 | C4≀C2 | C3×C4≀C2 |
| kernel | C3×C4≀C2 | C4×C12 | C3×M4(2) | C3×C4○D4 | C4≀C2 | C3×D4 | C3×Q8 | C42 | M4(2) | C4○D4 | D4 | Q8 | C12 | C2×C6 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 8 |
Matrix representation of C3×C4≀C2 ►in GL2(𝔽13) generated by
| 9 | 0 |
| 0 | 9 |
| 0 | 2 |
| 6 | 0 |
| 0 | 11 |
| 6 | 0 |
| 11 | 4 |
| 12 | 11 |
G:=sub<GL(2,GF(13))| [9,0,0,9],[0,6,2,0],[0,6,11,0],[11,12,4,11] >;
C3×C4≀C2 in GAP, Magma, Sage, TeX
C_3\times C_4\wr C_2
% in TeX
G:=Group("C3xC4wrC2"); // GroupNames label
G:=SmallGroup(96,54);
// by ID
G=gap.SmallGroup(96,54);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,729,117,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations
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