direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4×Q16, C42.453C23, C4.1402+ (1+4), C4⋊2(C2×Q16), C2.70(D42), (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×C4).484C24, (C2×C8).180C23, (C22×Q16)⋊13C2, C4.100(C22×D4), C23.468(C2×D4), C4⋊Q8.139C22, C2.18(C22×Q16), (C4×D4).327C22, (C4×Q8).145C22, (C2×Q8).392C23, (C2×Q16).36C22, 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
Subgroups: 424 in 236 conjugacy classes, 104 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×13], C22, C22 [×4], C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×20], D4 [×4], Q8 [×4], Q8 [×18], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], Q16 [×4], Q16 [×14], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×6], C2×Q8 [×16], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×4], C4⋊Q8 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16, C2×Q16 [×8], C2×Q16 [×8], C22×Q8 [×4], C8×D4, C4×Q16, C22⋊Q16 [×4], C4⋊2Q16 [×2], C8.18D4 [×2], C4⋊Q16, D4×Q8 [×2], C22×Q16 [×2], D4×Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C24, C2×Q16 [×6], C22×D4 [×2], 2+ (1+4), D42, C22×Q16, Q8○D8, D4×Q16
Generators and relations
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 >
►On 64 points
(1 16 19 48)(2 9 20 41)(3 10 21 42)(4 11 22 43)(5 12 23 44)(6 13 24 45)(7 14 17 46)(8 15 18 47)(25 34 52 60)(26 35 53 61)(27 36 54 62)(28 37 55 63)(29 38 56 64)(30 39 49 57)(31 40 50 58)(32 33 51 59)
(1 5)(2 6)(3 7)(4 8)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 36 13 40)(10 35 14 39)(11 34 15 38)(12 33 16 37)(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)| (1,16,19,48)(2,9,20,41)(3,10,21,42)(4,11,22,43)(5,12,23,44)(6,13,24,45)(7,14,17,46)(8,15,18,47)(25,34,52,60)(26,35,53,61)(27,36,54,62)(28,37,55,63)(29,38,56,64)(30,39,49,57)(31,40,50,58)(32,33,51,59), (1,5)(2,6)(3,7)(4,8)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(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( (1,16,19,48)(2,9,20,41)(3,10,21,42)(4,11,22,43)(5,12,23,44)(6,13,24,45)(7,14,17,46)(8,15,18,47)(25,34,52,60)(26,35,53,61)(27,36,54,62)(28,37,55,63)(29,38,56,64)(30,39,49,57)(31,40,50,58)(32,33,51,59), (1,5)(2,6)(3,7)(4,8)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(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([(1,16,19,48),(2,9,20,41),(3,10,21,42),(4,11,22,43),(5,12,23,44),(6,13,24,45),(7,14,17,46),(8,15,18,47),(25,34,52,60),(26,35,53,61),(27,36,54,62),(28,37,55,63),(29,38,56,64),(30,39,49,57),(31,40,50,58),(32,33,51,59)], [(1,5),(2,6),(3,7),(4,8),(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(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)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,36,13,40),(10,35,14,39),(11,34,15,38),(12,33,16,37),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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