direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4×Q16, C42.453C23, C4.1402+ 1+4, C2.70D42, C4⋊2(C2×Q16), (C4×Q16)⋊9C2, (C8×D4).8C2, C8.82(C2×D4), (D4×Q8).4C2, C4⋊C4.257D4, Q8.29(C2×D4), C22⋊2(C2×Q16), C4⋊Q16⋊12C2, C4⋊2Q16⋊13C2, (C2×D4).349D4, (C4×C8).81C22, C22⋊Q16⋊8C2, C22⋊C4.96D4, C2.43(Q8○D8), C8.18D4⋊10C2, C4⋊C4.225C23, C4⋊C8.295C22, (C2×C8).180C23, (C2×C4).484C24, (C22×Q16)⋊13C2, C4.100(C22×D4), C23.468(C2×D4), C4⋊Q8.139C22, C2.18(C22×Q16), (C4×D4).327C22, (C2×Q16).36C22, (C2×Q8).392C23, (C4×Q8).145C22, C2.D8.189C22, C22⋊Q8.68C22, C22⋊C8.183C22, (C22×C8).159C22, Q8⋊C4.10C22, C22.744(C22×D4), (C22×C4).1128C23, (C22×Q8).336C22, (C2×D4)○(C2×Q16), (C2×C4).162(C2×D4), SmallGroup(128,2018)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4×Q16
G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 424 in 236 conjugacy classes, 104 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×C8, C2×Q16, C2×Q16, C2×Q16, C22×Q8, C8×D4, C4×Q16, C22⋊Q16, C4⋊2Q16, C8.18D4, C4⋊Q16, D4×Q8, C22×Q16, D4×Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C22×D4, 2+ 1+4, D42, C22×Q16, Q8○D8, D4×Q16
►On 64 points
(1 34 19 48)(2 35 20 41)(3 36 21 42)(4 37 22 43)(5 38 23 44)(6 39 24 45)(7 40 17 46)(8 33 18 47)(9 50 58 25)(10 51 59 26)(11 52 60 27)(12 53 61 28)(13 54 62 29)(14 55 63 30)(15 56 64 31)(16 49 57 32)
(1 5)(2 6)(3 7)(4 8)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)(49 53)(50 54)(51 55)(52 56)
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
G:=sub<Sym(64)| (1,34,19,48)(2,35,20,41)(3,36,21,42)(4,37,22,43)(5,38,23,44)(6,39,24,45)(7,40,17,46)(8,33,18,47)(9,50,58,25)(10,51,59,26)(11,52,60,27)(12,53,61,28)(13,54,62,29)(14,55,63,30)(15,56,64,31)(16,49,57,32), (1,5)(2,6)(3,7)(4,8)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)(49,53)(50,54)(51,55)(52,56), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)>;
G:=Group( (1,34,19,48)(2,35,20,41)(3,36,21,42)(4,37,22,43)(5,38,23,44)(6,39,24,45)(7,40,17,46)(8,33,18,47)(9,50,58,25)(10,51,59,26)(11,52,60,27)(12,53,61,28)(13,54,62,29)(14,55,63,30)(15,56,64,31)(16,49,57,32), (1,5)(2,6)(3,7)(4,8)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)(49,53)(50,54)(51,55)(52,56), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63) );
G=PermutationGroup([[(1,34,19,48),(2,35,20,41),(3,36,21,42),(4,37,22,43),(5,38,23,44),(6,39,24,45),(7,40,17,46),(8,33,18,47),(9,50,58,25),(10,51,59,26),(11,52,60,27),(12,53,61,28),(13,54,62,29),(14,55,63,30),(15,56,64,31),(16,49,57,32)], [(1,5),(2,6),(3,7),(4,8),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42),(49,53),(50,54),(51,55),(52,56)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,30,5,26),(2,29,6,25),(3,28,7,32),(4,27,8,31),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 4L | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | Q16 | 2+ 1+4 | Q8○D8 |
| kernel | D4×Q16 | C8×D4 | C4×Q16 | C22⋊Q16 | C4⋊2Q16 | C8.18D4 | C4⋊Q16 | D4×Q8 | C22×Q16 | C22⋊C4 | C4⋊C4 | Q16 | C2×D4 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of D4×Q16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 10 | 16 | 0 | 0 |
| 16 | 7 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,1,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[10,16,0,0,16,7,0,0,0,0,16,0,0,0,0,16] >;
D4×Q16 in GAP, Magma, Sage, TeX
D_4\times Q_{16} % in TeX
G:=Group("D4xQ16"); // GroupNames label
G:=SmallGroup(128,2018);
// by ID
G=gap.SmallGroup(128,2018);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2019,346,248,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations