metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊5D8, D12⋊6D4, (C2×C8)⋊3D6, (C2×D8)⋊4S3, (C2×D4)⋊3D6, (C3×D4)⋊5D4, D6⋊C8⋊17C2, (C6×D8)⋊13C2, C6.45(C2×D8), C4.59(S3×D4), C2.28(S3×D8), D6⋊3D4⋊3C2, C3⋊4(C22⋊D8), D4⋊2(C3⋊D4), C12.46(C2×D4), (C6×D4)⋊3C22, (C2×C24)⋊19C22, C6.55C22≀C2, C2.D24⋊17C2, D4⋊Dic3⋊28C2, C2.29(D8⋊S3), C6.50(C8⋊C22), C4⋊Dic3⋊19C22, (C2×Dic3).65D4, (C22×S3).90D4, C22.256(S3×D4), C2.23(C23⋊2D6), (C2×C12).433C23, (C2×D12).116C22, (C2×S3×D4)⋊2C2, (C2×C3⋊C8)⋊7C22, (C2×D4⋊S3)⋊19C2, C4.36(C2×C3⋊D4), (C2×C6).346(C2×D4), (S3×C2×C4).44C22, (C2×C4).523(C22×S3), SmallGroup(192,715)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a4b, dcd=c-1 >
Subgroups: 760 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, 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, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C2×D8, C22×D4, C2×C3⋊C8, C4⋊Dic3, D4⋊S3, C6.D4, C2×C24, C3×D8, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C22⋊D8, D6⋊C8, C2.D24, D4⋊Dic3, C2×D4⋊S3, D6⋊3D4, C6×D8, C2×S3×D4, D12⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×D8, C8⋊C22, S3×D4, C2×C3⋊D4, C22⋊D8, S3×D8, D8⋊S3, C23⋊2D6, D12⋊D4
►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 17)(2 16)(3 15)(4 14)(5 13)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)
(1 30 15 37)(2 29 16 48)(3 28 17 47)(4 27 18 46)(5 26 19 45)(6 25 20 44)(7 36 21 43)(8 35 22 42)(9 34 23 41)(10 33 24 40)(11 32 13 39)(12 31 14 38)
(2 6)(3 11)(5 9)(8 12)(13 17)(14 22)(16 20)(19 23)(25 48)(26 41)(27 46)(28 39)(29 44)(30 37)(31 42)(32 47)(33 40)(34 45)(35 38)(36 43)
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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,48), (1,30,15,37)(2,29,16,48)(3,28,17,47)(4,27,18,46)(5,26,19,45)(6,25,20,44)(7,36,21,43)(8,35,22,42)(9,34,23,41)(10,33,24,40)(11,32,13,39)(12,31,14,38), (2,6)(3,11)(5,9)(8,12)(13,17)(14,22)(16,20)(19,23)(25,48)(26,41)(27,46)(28,39)(29,44)(30,37)(31,42)(32,47)(33,40)(34,45)(35,38)(36,43)>;
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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,48), (1,30,15,37)(2,29,16,48)(3,28,17,47)(4,27,18,46)(5,26,19,45)(6,25,20,44)(7,36,21,43)(8,35,22,42)(9,34,23,41)(10,33,24,40)(11,32,13,39)(12,31,14,38), (2,6)(3,11)(5,9)(8,12)(13,17)(14,22)(16,20)(19,23)(25,48)(26,41)(27,46)(28,39)(29,44)(30,37)(31,42)(32,47)(33,40)(34,45)(35,38)(36,43) );
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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(36,48)], [(1,30,15,37),(2,29,16,48),(3,28,17,47),(4,27,18,46),(5,26,19,45),(6,25,20,44),(7,36,21,43),(8,35,22,42),(9,34,23,41),(10,33,24,40),(11,32,13,39),(12,31,14,38)], [(2,6),(3,11),(5,9),(8,12),(13,17),(14,22),(16,20),(19,23),(25,48),(26,41),(27,46),(28,39),(29,44),(30,37),(31,42),(32,47),(33,40),(34,45),(35,38),(36,43)]])
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 | 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 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 2 | 2 | 2 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D8 | C3⋊D4 | C8⋊C22 | S3×D4 | S3×D4 | S3×D8 | D8⋊S3 |
| kernel | D12⋊D4 | D6⋊C8 | C2.D24 | D4⋊Dic3 | C2×D4⋊S3 | D6⋊3D4 | C6×D8 | C2×S3×D4 | C2×D8 | D12 | C2×Dic3 | C3×D4 | C22×S3 | 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 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12⋊D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 48 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 25 | 1 |
| 41 | 71 | 0 | 0 | 0 | 0 |
| 38 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 25 |
| 0 | 0 | 0 | 0 | 38 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 41 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,48,0,0,0,0,3,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,25,0,0,0,0,0,1],[41,38,0,0,0,0,71,32,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,38,0,0,0,0,25,0],[1,41,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D12⋊D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes D_4 % in TeX
G:=Group("D12:D4"); // GroupNames label
G:=SmallGroup(192,715);
// by ID
G=gap.SmallGroup(192,715);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^7*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations