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, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8○D4, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C4.Dic3, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D12.C4, D4.Dic3, S3×C3⋊C8, D6.Dic3, C3×C4.Dic3, C2×C32⋊4C8, C3×C4○D12, D12.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3, 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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 36)(12 35)(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 22 10 13 7 16 4 19)(2 17 11 20 8 23 5 14)(3 24 12 15 9 18 6 21)(25 37 28 46 31 43 34 40)(26 44 29 41 32 38 35 47)(27 39 30 48 33 45 36 42)
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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,36)(12,35)(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), (1,22,10,13,7,16,4,19)(2,17,11,20,8,23,5,14)(3,24,12,15,9,18,6,21)(25,37,28,46,31,43,34,40)(26,44,29,41,32,38,35,47)(27,39,30,48,33,45,36,42)>;
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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,36)(12,35)(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), (1,22,10,13,7,16,4,19)(2,17,11,20,8,23,5,14)(3,24,12,15,9,18,6,21)(25,37,28,46,31,43,34,40)(26,44,29,41,32,38,35,47)(27,39,30,48,33,45,36,42) );
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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,36),(12,35),(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)], [(1,22,10,13,7,16,4,19),(2,17,11,20,8,23,5,14),(3,24,12,15,9,18,6,21),(25,37,28,46,31,43,34,40),(26,44,29,41,32,38,35,47),(27,39,30,48,33,45,36,42)]])
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