metabelian, supersoluble, monomial
Aliases: D12.Dic3, Dic6.Dic3, C3⋊C8.21D6, C3⋊D4.Dic3, C12.31(C4×S3), C4○D12.3S3, (C4×S3).32D6, (C3×D12).2C4, C32⋊5(C8○D4), C4.Dic3⋊5S3, C4.9(S3×Dic3), C3⋊5(D12.C4), (C2×C12).108D6, C62.45(C2×C4), (C3×Dic6).2C4, D6.2(C2×Dic3), D6.Dic3⋊14C2, (S3×C12).9C22, C3⋊2(D4.Dic3), (C6×C12).67C22, C12.17(C2×Dic3), C22.2(S3×Dic3), C6.4(C22×Dic3), (C3×C12).144C23, C12.143(C22×S3), Dic3.2(C2×Dic3), C32⋊4C8.38C22, (S3×C3⋊C8)⋊13C2, C4.90(C2×S32), C6.84(S3×C2×C4), (C2×C4).105S32, C2.6(C2×S3×Dic3), (S3×C6).5(C2×C4), (C2×C6).17(C4×S3), (C3×C3⋊D4).2C4, (C3×C12).57(C2×C4), (C2×C32⋊4C8)⋊4C2, (C3×C4○D12).6C2, (C3×C3⋊C8).26C22, (C2×C6).8(C2×Dic3), (C3×C4.Dic3)⋊10C2, (C3×C6).40(C22×C4), (C3×Dic3).5(C2×C4), SmallGroup(288,463)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.Dic3
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a6c3, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c5 >
Subgroups: 338 in 135 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8 [×5], C4.Dic3, C4.Dic3 [×2], C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C32⋊4C8 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, D12.C4, D4.Dic3, S3×C3⋊C8 [×2], D6.Dic3 [×2], C3×C4.Dic3, C2×C32⋊4C8, C3×C4○D12, D12.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, D12.C4, D4.Dic3, C2×S3×Dic3, D12.Dic3
►On 48 points
(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 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)
(1 6 11 4 9 2 7 12 5 10 3 8)(13 20 15 22 17 24 19 14 21 16 23 18)(25 32 27 34 29 36 31 26 33 28 35 30)(37 42 47 40 45 38 43 48 41 46 39 44)
(1 13 10 16 7 19 4 22)(2 20 11 23 8 14 5 17)(3 15 12 18 9 21 6 24)(25 45 28 42 31 39 34 48)(26 40 29 37 32 46 35 43)(27 47 30 44 33 41 36 38)
G:=sub<Sym(48)| (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44), (1,6,11,4,9,2,7,12,5,10,3,8)(13,20,15,22,17,24,19,14,21,16,23,18)(25,32,27,34,29,36,31,26,33,28,35,30)(37,42,47,40,45,38,43,48,41,46,39,44), (1,13,10,16,7,19,4,22)(2,20,11,23,8,14,5,17)(3,15,12,18,9,21,6,24)(25,45,28,42,31,39,34,48)(26,40,29,37,32,46,35,43)(27,47,30,44,33,41,36,38)>;
G:=Group( (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44), (1,6,11,4,9,2,7,12,5,10,3,8)(13,20,15,22,17,24,19,14,21,16,23,18)(25,32,27,34,29,36,31,26,33,28,35,30)(37,42,47,40,45,38,43,48,41,46,39,44), (1,13,10,16,7,19,4,22)(2,20,11,23,8,14,5,17)(3,15,12,18,9,21,6,24)(25,45,28,42,31,39,34,48)(26,40,29,37,32,46,35,43)(27,47,30,44,33,41,36,38) );
G=PermutationGroup([(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,48),(21,47),(22,46),(23,45),(24,44)], [(1,6,11,4,9,2,7,12,5,10,3,8),(13,20,15,22,17,24,19,14,21,16,23,18),(25,32,27,34,29,36,31,26,33,28,35,30),(37,42,47,40,45,38,43,48,41,46,39,44)], [(1,13,10,16,7,19,4,22),(2,20,11,23,8,14,5,17),(3,15,12,18,9,21,6,24),(25,45,28,42,31,39,34,48),(26,40,29,37,32,46,35,43),(27,47,30,44,33,41,36,38)])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 4 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
48 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 | C4 | C4 | C4 | S3 | S3 | D6 | Dic3 | D6 | Dic3 | Dic3 | D6 | C4×S3 | C4×S3 | C8○D4 | S32 | S3×Dic3 | C2×S32 | S3×Dic3 | D12.C4 | D4.Dic3 | D12.Dic3 |
| kernel | D12.Dic3 | S3×C3⋊C8 | D6.Dic3 | C3×C4.Dic3 | C2×C32⋊4C8 | C3×C4○D12 | C3×Dic6 | C3×D12 | C3×C3⋊D4 | C4.Dic3 | C4○D12 | C3⋊C8 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of D12.Dic3 ►in GL4(𝔽5) generated by
| 0 | 1 | 1 | 4 |
| 2 | 3 | 2 | 3 |
| 2 | 2 | 0 | 1 |
| 1 | 1 | 2 | 2 |
| 2 | 4 | 0 | 2 |
| 2 | 3 | 1 | 0 |
| 0 | 4 | 2 | 1 |
| 2 | 0 | 3 | 3 |
| 4 | 0 | 1 | 0 |
| 0 | 2 | 0 | 3 |
| 2 | 0 | 4 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 3 | 0 | 4 |
| 1 | 0 | 2 | 0 |
| 0 | 2 | 0 | 3 |
| 1 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [0,2,2,1,1,3,2,1,1,2,0,2,4,3,1,2],[2,2,0,2,4,3,4,0,0,1,2,3,2,0,1,3],[4,0,2,0,0,2,0,1,1,0,4,0,0,3,0,1],[0,1,0,1,3,0,2,0,0,2,0,1,4,0,3,0] >;
D12.Dic3 in GAP, Magma, Sage, TeX
D_{12}.{\rm Dic}_3 % in TeX
G:=Group("D12.Dic3"); // GroupNames label
G:=SmallGroup(288,463);
// by ID
G=gap.SmallGroup(288,463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,219,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^6*c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^5>;
// generators/relations