direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D12, C42⋊15D6, C6.1022+ (1+4), C3⋊1(D42), C4⋊4(S3×D4), C4⋊C4⋊46D6, D6⋊5(C2×D4), (C3×D4)⋊9D4, C12⋊1(C2×D4), C4⋊1(C2×D12), (C4×D4)⋊11S3, D6⋊D4⋊5C2, (D4×C12)⋊13C2, (C4×D12)⋊27C2, C12⋊7D4⋊7C2, D6⋊C4⋊4C22, C22⋊C4⋊45D6, C22⋊2(C2×D12), (C22×C4)⋊14D6, C12⋊D4⋊14C2, C4⋊D12⋊11C2, (C4×C12)⋊18C22, (C2×D4).247D6, (C22×D12)⋊8C2, (C2×C6).93C24, C6.15(C22×D4), C2.14(D4○D12), (C2×D12)⋊16C22, (S3×C23)⋊5C22, C4⋊Dic3⋊58C22, (C22×C12)⋊9C22, C2.17(C22×D12), (C2×C12).158C23, (C6×D4).256C22, (C22×C6).163C23, C23.181(C22×S3), C22.118(S3×C23), (C2×Dic3).39C23, (C22×S3).171C23, (C2×S3×D4)⋊3C2, (C2×C6)⋊1(C2×D4), C2.21(C2×S3×D4), (S3×C2×C4)⋊2C22, (C3×C4⋊C4)⋊58C22, (C2×C3⋊D4)⋊2C22, (C3×C22⋊C4)⋊49C22, (C2×C4).157(C22×S3), SmallGroup(192,1108)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1432 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×4], C4 [×5], C22, C22 [×4], C22 [×40], S3 [×8], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×4], D4 [×30], C23 [×2], C23 [×26], Dic3 [×2], C12 [×4], C12 [×3], D6 [×4], D6 [×32], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×31], C24 [×4], C4×S3 [×4], D12 [×4], D12 [×18], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×S3 [×6], C22×S3 [×20], C22×C6 [×2], C4×D4, C4×D4, C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4⋊Dic3, D6⋊C4 [×6], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×10], C2×D12 [×8], S3×D4 [×8], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23 [×4], D42, C4×D12, C4⋊D12, D6⋊D4 [×4], C12⋊D4 [×2], C12⋊7D4 [×2], D4×C12, C22×D12 [×2], C2×S3×D4 [×2], D4×D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C24, D12 [×4], C22×S3 [×7], C22×D4 [×2], 2+ (1+4), C2×D12 [×6], S3×D4 [×2], S3×C23, D42, C22×D12, C2×S3×D4, D4○D12, D4×D12
Generators and relations
G = < a,b,c,d | a4=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 48 points
(1 41 33 19)(2 42 34 20)(3 43 35 21)(4 44 36 22)(5 45 25 23)(6 46 26 24)(7 47 27 13)(8 48 28 14)(9 37 29 15)(10 38 30 16)(11 39 31 17)(12 40 32 18)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(37 47)(38 46)(39 45)(40 44)(41 43)
G:=sub<Sym(48)| (1,41,33,19)(2,42,34,20)(3,43,35,21)(4,44,36,22)(5,45,25,23)(6,46,26,24)(7,47,27,13)(8,48,28,14)(9,37,29,15)(10,38,30,16)(11,39,31,17)(12,40,32,18), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,47)(38,46)(39,45)(40,44)(41,43)>;
G:=Group( (1,41,33,19)(2,42,34,20)(3,43,35,21)(4,44,36,22)(5,45,25,23)(6,46,26,24)(7,47,27,13)(8,48,28,14)(9,37,29,15)(10,38,30,16)(11,39,31,17)(12,40,32,18), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,47)(38,46)(39,45)(40,44)(41,43) );
G=PermutationGroup([(1,41,33,19),(2,42,34,20),(3,43,35,21),(4,44,36,22),(5,45,25,23),(6,46,26,24),(7,47,27,13),(8,48,28,14),(9,37,29,15),(10,38,30,16),(11,39,31,17),(12,40,32,18)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21),(25,31),(26,30),(27,29),(32,36),(33,35),(37,47),(38,46),(39,45),(40,44),(41,43)])
Matrix representation ►G ⊆ GL6(ℤ)
| 1 | -2 | 0 | 0 | 0 | 0 |
| 1 | -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 | -2 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -2 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,Integers())| [1,1,0,0,0,0,-2,-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],[1,0,0,0,0,0,-2,-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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | D12 | 2+ (1+4) | S3×D4 | D4○D12 |
| kernel | D4×D12 | C4×D12 | C4⋊D12 | D6⋊D4 | C12⋊D4 | C12⋊7D4 | D4×C12 | C22×D12 | C2×S3×D4 | C4×D4 | D12 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_4\times D_{12} % in TeX
G:=Group("D4xD12"); // GroupNames label
G:=SmallGroup(192,1108);
// by ID
G=gap.SmallGroup(192,1108);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations