metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊16D4, C4⋊C4⋊3D6, (C2×C6)⋊2D8, (C2×D4)⋊1D6, C4⋊D4⋊1S3, C4.98(S3×D4), C6.54(C2×D8), C3⋊3(C22⋊D8), (C2×C12).71D4, (C6×D4)⋊1C22, C6.44C22≀C2, C22⋊3(D4⋊S3), C6.D8⋊33C2, C12.145(C2×D4), (C22×C6).82D4, (C22×D12)⋊13C2, (C22×C4).135D6, C2.12(C23⋊2D6), C2.12(D4⋊D6), C12.55D4⋊10C2, C6.114(C8⋊C22), (C2×C12).355C23, C23.65(C3⋊D4), (C2×D12).240C22, (C22×C12).159C22, (C2×D4⋊S3)⋊8C2, (C2×C3⋊C8)⋊5C22, C2.9(C2×D4⋊S3), (C3×C4⋊D4)⋊1C2, (C3×C4⋊C4)⋊5C22, (C2×C6).486(C2×D4), (C2×C4).49(C3⋊D4), (C2×C4).455(C22×S3), C22.161(C2×C3⋊D4), SmallGroup(192,595)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊16D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd=c-1 >
Subgroups: 752 in 198 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C2×C3⋊C8, D4⋊S3, C3×C22⋊C4, C3×C4⋊C4, C2×D12, C2×D12, C22×C12, C6×D4, C6×D4, S3×C23, C22⋊D8, C6.D8, C12.55D4, C2×D4⋊S3, C3×C4⋊D4, C22×D12, D12⋊16D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×D8, C8⋊C22, D4⋊S3, S3×D4, C2×C3⋊D4, C22⋊D8, C2×D4⋊S3, C23⋊2D6, D4⋊D6, D12⋊16D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 24)(10 23)(11 22)(12 21)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 48)(33 47)(34 46)(35 45)(36 44)
(1 25 21 47)(2 32 22 42)(3 27 23 37)(4 34 24 44)(5 29 13 39)(6 36 14 46)(7 31 15 41)(8 26 16 48)(9 33 17 43)(10 28 18 38)(11 35 19 45)(12 30 20 40)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,24)(10,23)(11,22)(12,21)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44), (1,25,21,47)(2,32,22,42)(3,27,23,37)(4,34,24,44)(5,29,13,39)(6,36,14,46)(7,31,15,41)(8,26,16,48)(9,33,17,43)(10,28,18,38)(11,35,19,45)(12,30,20,40), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,24)(10,23)(11,22)(12,21)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44), (1,25,21,47)(2,32,22,42)(3,27,23,37)(4,34,24,44)(5,29,13,39)(6,36,14,46)(7,31,15,41)(8,26,16,48)(9,33,17,43)(10,28,18,38)(11,35,19,45)(12,30,20,40), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,24),(10,23),(11,22),(12,21),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,48),(33,47),(34,46),(35,45),(36,44)], [(1,25,21,47),(2,32,22,42),(3,27,23,37),(4,34,24,44),(5,29,13,39),(6,36,14,46),(7,31,15,41),(8,26,16,48),(9,33,17,43),(10,28,18,38),(11,35,19,45),(12,30,20,40)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40)]])
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 | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D8 | C3⋊D4 | C3⋊D4 | C8⋊C22 | S3×D4 | D4⋊S3 | D4⋊D6 |
| kernel | D12⋊16D4 | C6.D8 | C12.55D4 | C2×D4⋊S3 | C3×C4⋊D4 | C22×D12 | C4⋊D4 | D12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of D12⋊16D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 71 | 0 | 0 | 0 | 0 |
| 37 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,37,0,0,0,0,71,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,57,0,0,0,0,57,16],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
D12⋊16D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{16}D_4 % in TeX
G:=Group("D12:16D4"); // GroupNames label
G:=SmallGroup(192,595);
// by ID
G=gap.SmallGroup(192,595);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations