metabelian, supersoluble, monomial
Aliases: Dic3⋊4D12, C62.50C23, D6⋊1(C4×S3), C3⋊2(C4×D12), C32⋊3(C4×D4), D6⋊C4⋊20S3, C2.1(S3×D12), C6.12(S3×D4), C3⋊D12⋊1C4, (C3×Dic3)⋊7D4, Dic3⋊3(C4×S3), C6.12(C2×D12), (Dic3×C12)⋊2C2, (C4×Dic3)⋊15S3, (C2×C12).197D6, C6.53(C4○D12), (C22×S3).32D6, Dic3⋊Dic3⋊13C2, C3⋊1(Dic3⋊4D4), C6.40(D4⋊2S3), C6.11D12⋊15C2, (C6×C12).228C22, (C2×Dic3).111D6, C2.4(D6.3D6), (C6×Dic3).153C22, C2.16(C4×S32), (C2×C4).48S32, C6.15(S3×C2×C4), (S3×C6)⋊2(C2×C4), (C2×S3×Dic3)⋊8C2, (C3×D6⋊C4)⋊21C2, C22.30(C2×S32), (C3×C6).43(C2×D4), (C3×Dic3)⋊7(C2×C4), (C2×C6.D6)⋊8C2, (S3×C2×C6).13C22, (C2×C3⋊D12).5C2, (C3×C6).64(C4○D4), (C2×C6).69(C22×S3), (C3×C6).14(C22×C4), (C22×C3⋊S3).13C22, (C2×C3⋊Dic3).37C22, (C2×C3⋊S3)⋊1(C2×C4), SmallGroup(288,528)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊4D12
G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 826 in 205 conjugacy classes, 62 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C22⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3, C6.D6, C3⋊D12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C4×D12, Dic3⋊4D4, Dic3⋊Dic3, Dic3×C12, C3×D6⋊C4, C6.11D12, C2×S3×Dic3, C2×C6.D6, C2×C3⋊D12, Dic3⋊4D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, S32, S3×C2×C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×S32, C4×D12, Dic3⋊4D4, C4×S32, S3×D12, D6.3D6, Dic3⋊4D12
►On 48 points
(1 23 9 19 5 15)(2 16 6 20 10 24)(3 13 11 21 7 17)(4 18 8 22 12 14)(25 39 29 43 33 47)(26 48 34 44 30 40)(27 41 31 45 35 37)(28 38 36 46 32 42)
(1 27 19 45)(2 28 20 46)(3 29 21 47)(4 30 22 48)(5 31 23 37)(6 32 24 38)(7 33 13 39)(8 34 14 40)(9 35 15 41)(10 36 16 42)(11 25 17 43)(12 26 18 44)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(37 43)(38 42)(39 41)(44 48)(45 47)
G:=sub<Sym(48)| (1,23,9,19,5,15)(2,16,6,20,10,24)(3,13,11,21,7,17)(4,18,8,22,12,14)(25,39,29,43,33,47)(26,48,34,44,30,40)(27,41,31,45,35,37)(28,38,36,46,32,42), (1,27,19,45)(2,28,20,46)(3,29,21,47)(4,30,22,48)(5,31,23,37)(6,32,24,38)(7,33,13,39)(8,34,14,40)(9,35,15,41)(10,36,16,42)(11,25,17,43)(12,26,18,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,43)(38,42)(39,41)(44,48)(45,47)>;
G:=Group( (1,23,9,19,5,15)(2,16,6,20,10,24)(3,13,11,21,7,17)(4,18,8,22,12,14)(25,39,29,43,33,47)(26,48,34,44,30,40)(27,41,31,45,35,37)(28,38,36,46,32,42), (1,27,19,45)(2,28,20,46)(3,29,21,47)(4,30,22,48)(5,31,23,37)(6,32,24,38)(7,33,13,39)(8,34,14,40)(9,35,15,41)(10,36,16,42)(11,25,17,43)(12,26,18,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,43)(38,42)(39,41)(44,48)(45,47) );
G=PermutationGroup([[(1,23,9,19,5,15),(2,16,6,20,10,24),(3,13,11,21,7,17),(4,18,8,22,12,14),(25,39,29,43,33,47),(26,48,34,44,30,40),(27,41,31,45,35,37),(28,38,36,46,32,42)], [(1,27,19,45),(2,28,20,46),(3,29,21,47),(4,30,22,48),(5,31,23,37),(6,32,24,38),(7,33,13,39),(8,34,14,40),(9,35,15,41),(10,36,16,42),(11,25,17,43),(12,26,18,44)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21),(25,31),(26,30),(27,29),(32,36),(33,35),(37,43),(38,42),(39,41),(44,48),(45,47)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R | 12S | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | D12 | C4×S3 | C4○D12 | S32 | S3×D4 | D4⋊2S3 | C2×S32 | C4×S32 | S3×D12 | D6.3D6 |
| kernel | Dic3⋊4D12 | Dic3⋊Dic3 | Dic3×C12 | C3×D6⋊C4 | C6.11D12 | C2×S3×Dic3 | C2×C6.D6 | C2×C3⋊D12 | C3⋊D12 | C4×Dic3 | D6⋊C4 | C3×Dic3 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | Dic3 | D6 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 3 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of Dic3⋊4D12 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,8,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
Dic3⋊4D12 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_4D_{12} % in TeX
G:=Group("Dic3:4D12"); // GroupNames label
G:=SmallGroup(288,528);
// by ID
G=gap.SmallGroup(288,528);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations