direct product, metabelian, supersoluble, monomial
Aliases: C3×C4○D12, D12⋊5C6, Dic6⋊5C6, C12.68D6, C62.29C22, (C4×S3)⋊4C6, (C2×C12)⋊7S3, (C6×C12)⋊6C2, (C2×C12)⋊4C6, C3⋊D4⋊3C6, (S3×C12)⋊9C2, C4.16(S3×C6), D6.1(C2×C6), (C2×C6).18D6, (C3×D12)⋊11C2, C12.12(C2×C6), C32⋊7(C4○D4), C22.2(S3×C6), C6.4(C22×C6), (C3×Dic6)⋊11C2, C6.43(C22×S3), (C3×C6).22C23, Dic3.2(C2×C6), (S3×C6).10C22, (C3×C12).41C22, (C3×Dic3).11C22, C2.5(S3×C2×C6), (C2×C4)⋊3(C3×S3), C3⋊1(C3×C4○D4), (C3×C3⋊D4)⋊7C2, (C2×C6).14(C2×C6), SmallGroup(144,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4○D12
G = < a,b,c,d | a3=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
Subgroups: 168 in 88 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C3×C4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, C4○D12, C3×C4○D4, S3×C2×C6, C3×C4○D12
►On 24 points - transitive group 24T210
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)]])
G:=TransitiveGroup(24,210);
C3×C4○D12 is a maximal subgroup of
D12⋊4Dic3 D12⋊2Dic3 D12.2Dic3 D12.Dic3 D12.30D6 D12⋊20D6 D12⋊18D6 D12.32D6 D12.27D6 D12.29D6 D12.33D6 D12.34D6 D12⋊23D6 D12⋊24D6 D12⋊27D6 C3×S3×C4○D4 C62.36D6 D36⋊6C6 C62.47D6
C3×C4○D12 is a maximal quotient of
C12×Dic6 C12×D12 C12×C3⋊D4 C62.36D6 D36⋊6C6
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6M | 6N | 6O | 6P | 6Q | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 12S | 12T | 12U | 12V |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 |
| kernel | C3×C4○D12 | C3×Dic6 | S3×C12 | C3×D12 | C3×C3⋊D4 | C6×C12 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C2×C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C4○D12 ►in GL2(𝔽13) generated by
| 3 | 0 |
| 0 | 3 |
| 5 | 0 |
| 0 | 5 |
| 6 | 11 |
| 11 | 3 |
| 10 | 4 |
| 11 | 3 |
G:=sub<GL(2,GF(13))| [3,0,0,3],[5,0,0,5],[6,11,11,3],[10,11,4,3] >;
C3×C4○D12 in GAP, Magma, Sage, TeX
C_3\times C_4\circ D_{12} % in TeX
G:=Group("C3xC4oD12"); // GroupNames label
G:=SmallGroup(144,161);
// by ID
G=gap.SmallGroup(144,161);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations