p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊13D4, C42.454C23, C4.1412+ 1+4, C2.71D42, C4⋊C4○2Q16, (C8×D4)⋊15C2, C4⋊4(C4○D8), 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), (C2×D4).218C23, (C4×D4).328C22, C4⋊D4.69C22, C4⋊1D4.80C22, (C4×Q8).146C22, (C2×Q8).393C23, 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
Generators and relations for Q16⋊13D4
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 >
Subgroups: 552 in 252 conjugacy classes, 96 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4⋊1D4, C4⋊1D4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C8×D4, C4×Q16, D4⋊D4, C4⋊SD16, C8⋊7D4, C8⋊4D4, Q8⋊6D4, C2×C4○D8, Q16⋊13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2+ 1+4, D42, C2×C4○D8, D4○D8, Q16⋊13D4
►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 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)
(1 35 23 45)(2 34 24 44)(3 33 17 43)(4 40 18 42)(5 39 19 41)(6 38 20 48)(7 37 21 47)(8 36 22 46)(9 55 62 26)(10 54 63 25)(11 53 64 32)(12 52 57 31)(13 51 58 30)(14 50 59 29)(15 49 60 28)(16 56 61 27)
(1 8)(2 7)(3 6)(4 5)(9 57)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 20)(18 19)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(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), (1,30,5,26)(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), (1,35,23,45)(2,34,24,44)(3,33,17,43)(4,40,18,42)(5,39,19,41)(6,38,20,48)(7,37,21,47)(8,36,22,46)(9,55,62,26)(10,54,63,25)(11,53,64,32)(12,52,57,31)(13,51,58,30)(14,50,59,29)(15,49,60,28)(16,56,61,27), (1,8)(2,7)(3,6)(4,5)(9,57)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,20)(18,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(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), (1,30,5,26)(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), (1,35,23,45)(2,34,24,44)(3,33,17,43)(4,40,18,42)(5,39,19,41)(6,38,20,48)(7,37,21,47)(8,36,22,46)(9,55,62,26)(10,54,63,25)(11,53,64,32)(12,52,57,31)(13,51,58,30)(14,50,59,29)(15,49,60,28)(16,56,61,27), (1,8)(2,7)(3,6)(4,5)(9,57)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,20)(18,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(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)], [(1,30,5,26),(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)], [(1,35,23,45),(2,34,24,44),(3,33,17,43),(4,40,18,42),(5,39,19,41),(6,38,20,48),(7,37,21,47),(8,36,22,46),(9,55,62,26),(10,54,63,25),(11,53,64,32),(12,52,57,31),(13,51,58,30),(14,50,59,29),(15,49,60,28),(16,56,61,27)], [(1,8),(2,7),(3,6),(4,5),(9,57),(10,64),(11,63),(12,62),(13,61),(14,60),(15,59),(16,58),(17,20),(18,19),(21,24),(22,23),(25,32),(26,31),(27,30),(28,29),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,50),(51,56),(52,55),(53,54)]])
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 |
Matrix representation of Q16⋊13D4 ►in 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] >;
Q16⋊13D4 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