p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊13D4, C42.454C23, C4.1412+ (1+4), C4⋊C4○2Q16, (C8×D4)⋊15C2, C4⋊4(C4○D8), C2.71(D42), C8.83(C2×D4), D4⋊D4⋊8C2, C8⋊4D4⋊12C2, C8⋊7D4⋊22C2, C4⋊C4.407D4, (C4×Q16)⋊10C2, Q8⋊6D4⋊6C2, Q8.30(C2×D4), C4⋊SD16⋊41C2, (C2×D4).232D4, C2.43(D4○D8), (C4×C8).82C22, C22⋊C4.97D4, C4⋊C8.342C22, C4⋊C4.226C23, (C2×C4).485C24, (C2×C8).570C23, (C2×D8).35C22, C23.105(C2×D4), C4.101(C22×D4), (C4×D4).328C22, (C2×D4).218C23, C4⋊1D4.80C22, C4⋊D4.69C22, (C2×Q8).393C23, (C4×Q8).146C22, C2.D8.190C22, C22⋊C8.200C22, (C22×C8).195C22, (C2×Q16).171C22, C22.745(C22×D4), D4⋊C4.117C22, (C22×C4).1129C23, Q8⋊C4.158C22, (C2×SD16).153C22, C4⋊C4○(C2×Q16), (C2×C4○D8)⋊14C2, C2.57(C2×C4○D8), (C2×C4).922(C2×D4), (C2×C4○D4).195C22, SmallGroup(128,2019)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 552 in 252 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×18], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×28], Q8 [×4], Q8 [×2], C23 [×2], C23 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×6], SD16 [×8], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×12], C4×C8, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×4], C4⋊D4 [×4], C4⋊1D4 [×2], C4⋊1D4 [×2], C22×C8 [×2], C2×D8 [×4], C2×SD16 [×4], C2×Q16, C4○D8 [×8], C2×C4○D4 [×4], C8×D4, C4×Q16, D4⋊D4 [×4], C4⋊SD16 [×2], C8⋊7D4 [×2], C8⋊4D4, Q8⋊6D4 [×2], C2×C4○D8 [×2], Q16⋊13D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C4○D8 [×2], C22×D4 [×2], 2+ (1+4), D42, C2×C4○D8, D4○D8, Q16⋊13D4
Generators and relations
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=dad=a-1, bc=cb, dbd=a4b, dcd=c-1 >
►On 64 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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 10 5 14)(2 9 6 13)(3 16 7 12)(4 15 8 11)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 38 29 34)(26 37 30 33)(27 36 31 40)(28 35 32 39)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 27 23 45)(2 26 24 44)(3 25 17 43)(4 32 18 42)(5 31 19 41)(6 30 20 48)(7 29 21 47)(8 28 22 46)(9 37 50 59)(10 36 51 58)(11 35 52 57)(12 34 53 64)(13 33 54 63)(14 40 55 62)(15 39 56 61)(16 38 49 60)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 20)(18 19)(21 24)(22 23)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(49 50)(51 56)(52 55)(53 54)
G:=sub<Sym(64)| (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), (1,10,5,14)(2,9,6,13)(3,16,7,12)(4,15,8,11)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,38,29,34)(26,37,30,33)(27,36,31,40)(28,35,32,39)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63), (1,27,23,45)(2,26,24,44)(3,25,17,43)(4,32,18,42)(5,31,19,41)(6,30,20,48)(7,29,21,47)(8,28,22,46)(9,37,50,59)(10,36,51,58)(11,35,52,57)(12,34,53,64)(13,33,54,63)(14,40,55,62)(15,39,56,61)(16,38,49,60), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,20)(18,19)(21,24)(22,23)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(49,50)(51,56)(52,55)(53,54)>;
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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,10,5,14)(2,9,6,13)(3,16,7,12)(4,15,8,11)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,38,29,34)(26,37,30,33)(27,36,31,40)(28,35,32,39)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63), (1,27,23,45)(2,26,24,44)(3,25,17,43)(4,32,18,42)(5,31,19,41)(6,30,20,48)(7,29,21,47)(8,28,22,46)(9,37,50,59)(10,36,51,58)(11,35,52,57)(12,34,53,64)(13,33,54,63)(14,40,55,62)(15,39,56,61)(16,38,49,60), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,20)(18,19)(21,24)(22,23)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(49,50)(51,56)(52,55)(53,54) );
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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,10,5,14),(2,9,6,13),(3,16,7,12),(4,15,8,11),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(25,38,29,34),(26,37,30,33),(27,36,31,40),(28,35,32,39),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63)], [(1,27,23,45),(2,26,24,44),(3,25,17,43),(4,32,18,42),(5,31,19,41),(6,30,20,48),(7,29,21,47),(8,28,22,46),(9,37,50,59),(10,36,51,58),(11,35,52,57),(12,34,53,64),(13,33,54,63),(14,40,55,62),(15,39,56,61),(16,38,49,60)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,20),(18,19),(21,24),(22,23),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,64),(34,63),(35,62),(36,61),(37,60),(38,59),(39,58),(40,57),(49,50),(51,56),(52,55),(53,54)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 3 | 3 | 0 | 0 |
| 14 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 |
| 12 | 5 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 5 | 5 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 13 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 13 | 16 |
G:=sub<GL(4,GF(17))| [3,14,0,0,3,3,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,5,12,0,0,0,0,4,0,0,0,2,13],[14,3,0,0,3,3,0,0,0,0,1,13,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | C4○D8 | 2+ (1+4) | D4○D8 |
| kernel | Q16⋊13D4 | C8×D4 | C4×Q16 | D4⋊D4 | C4⋊SD16 | C8⋊7D4 | C8⋊4D4 | Q8⋊6D4 | C2×C4○D8 | C22⋊C4 | C4⋊C4 | Q16 | C2×D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_{16}\rtimes_{13}D_4 % in TeX
G:=Group("Q16:13D4"); // GroupNames label
G:=SmallGroup(128,2019);
// by ID
G=gap.SmallGroup(128,2019);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,2019,346,248,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations