metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊4D8, D4⋊2D12, D12⋊2D4, C4⋊C4⋊1D6, (C2×C8)⋊2D6, D6⋊C8⋊4C2, (C3×D4)⋊1D4, C2.8(S3×D8), (C2×D24)⋊4C2, C4.1(C2×D12), C4.84(S3×D4), C6.22(C2×D8), C12⋊D4⋊1C2, D4⋊C4⋊4S3, C3⋊2(C22⋊D8), (C2×C24)⋊2C22, C6.D8⋊6C2, C6.19C22≀C2, (C2×D4).134D6, C12.107(C2×D4), C2.10(Q8⋊3D6), (C2×D12)⋊12C22, C6.55(C8⋊C22), (C2×Dic3).18D4, (C6×D4).34C22, (C22×S3).70D4, C22.171(S3×D4), C2.22(D6⋊D4), (C2×C12).213C23, (C2×S3×D4)⋊1C2, (C2×D4⋊S3)⋊2C2, (C2×C3⋊C8)⋊2C22, (C3×C4⋊C4)⋊3C22, (S3×C2×C4).7C22, (C3×D4⋊C4)⋊4C2, (C2×C6).226(C2×D4), (C2×C4).320(C22×S3), SmallGroup(192,332)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D12
G = < a,b,c,d | a4=b2=c12=d2=1, bab=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >
Subgroups: 808 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, D4⋊C4, C4⋊D4, C2×D8, C22×D4, D24, C2×C3⋊C8, D6⋊C4, D4⋊S3, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C22⋊D8, C6.D8, D6⋊C8, C3×D4⋊C4, C12⋊D4, C2×D24, C2×D4⋊S3, C2×S3×D4, D4⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C22≀C2, C2×D8, C8⋊C22, C2×D12, S3×D4, C22⋊D8, D6⋊D4, S3×D8, Q8⋊3D6, D4⋊D12
►On 48 points
(1 30 20 41)(2 42 21 31)(3 32 22 43)(4 44 23 33)(5 34 24 45)(6 46 13 35)(7 36 14 47)(8 48 15 25)(9 26 16 37)(10 38 17 27)(11 28 18 39)(12 40 19 29)
(1 47)(2 8)(3 37)(4 10)(5 39)(6 12)(7 41)(9 43)(11 45)(13 19)(14 30)(15 21)(16 32)(17 23)(18 34)(20 36)(22 26)(24 28)(25 42)(27 44)(29 46)(31 48)(33 38)(35 40)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 24)(9 23)(10 22)(11 21)(12 20)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
G:=sub<Sym(48)| (1,30,20,41)(2,42,21,31)(3,32,22,43)(4,44,23,33)(5,34,24,45)(6,46,13,35)(7,36,14,47)(8,48,15,25)(9,26,16,37)(10,38,17,27)(11,28,18,39)(12,40,19,29), (1,47)(2,8)(3,37)(4,10)(5,39)(6,12)(7,41)(9,43)(11,45)(13,19)(14,30)(15,21)(16,32)(17,23)(18,34)(20,36)(22,26)(24,28)(25,42)(27,44)(29,46)(31,48)(33,38)(35,40), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,24)(9,23)(10,22)(11,21)(12,20)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46)>;
G:=Group( (1,30,20,41)(2,42,21,31)(3,32,22,43)(4,44,23,33)(5,34,24,45)(6,46,13,35)(7,36,14,47)(8,48,15,25)(9,26,16,37)(10,38,17,27)(11,28,18,39)(12,40,19,29), (1,47)(2,8)(3,37)(4,10)(5,39)(6,12)(7,41)(9,43)(11,45)(13,19)(14,30)(15,21)(16,32)(17,23)(18,34)(20,36)(22,26)(24,28)(25,42)(27,44)(29,46)(31,48)(33,38)(35,40), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,24)(9,23)(10,22)(11,21)(12,20)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46) );
G=PermutationGroup([[(1,30,20,41),(2,42,21,31),(3,32,22,43),(4,44,23,33),(5,34,24,45),(6,46,13,35),(7,36,14,47),(8,48,15,25),(9,26,16,37),(10,38,17,27),(11,28,18,39),(12,40,19,29)], [(1,47),(2,8),(3,37),(4,10),(5,39),(6,12),(7,41),(9,43),(11,45),(13,19),(14,30),(15,21),(16,32),(17,23),(18,34),(20,36),(22,26),(24,28),(25,42),(27,44),(29,46),(31,48),(33,38),(35,40)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,24),(9,23),(10,22),(11,21),(12,20),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | D8 | D12 | C8⋊C22 | S3×D4 | S3×D4 | S3×D8 | Q8⋊3D6 |
| kernel | D4⋊D12 | C6.D8 | D6⋊C8 | C3×D4⋊C4 | C12⋊D4 | C2×D24 | C2×D4⋊S3 | C2×S3×D4 | D4⋊C4 | D12 | C2×Dic3 | C3×D4 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | D6 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D4⋊D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 41 |
| 0 | 0 | 0 | 0 | 16 | 41 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 32 |
| 0 | 0 | 0 | 0 | 57 | 32 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,0,0,0,0,71,1,0,0,0,0,0,0,32,16,0,0,0,0,41,41],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,71,1,0,0,0,0,0,0,41,57,0,0,0,0,32,32] >;
D4⋊D12 in GAP, Magma, Sage, TeX
D_4\rtimes D_{12} % in TeX
G:=Group("D4:D12"); // GroupNames label
G:=SmallGroup(192,332);
// by ID
G=gap.SmallGroup(192,332);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations