p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16:4D4, C42.444C23, C4.1312+ 1+4, C2.61D42, C8.9(C2xD4), C8:6D4:8C2, C8:D4:32C2, C4:C4.152D4, Q8:5D4:5C2, (C4xQ16):31C2, Q8.24(C2xD4), D4:D4:40C2, (C2xD4).166D4, C4:C8.96C22, (C2xC8).91C23, C2.40(Q8oD8), C4.91(C22xD4), C8.12D4:23C2, C4:C4.216C23, (C2xC4).475C24, (C4xC8).190C22, Q8.D4:39C2, C22:Q16:29C2, C22:C4.162D4, (C2xD8).33C22, C23.317(C2xD4), (C2xD4).213C23, (C4xD4).149C22, C4:D4.63C22, C22:C8.74C22, (C2xQ16).34C22, (C2xQ8).390C23, (C4xQ8).141C22, C2.D8.222C22, C22:Q8.62C22, D4:C4.67C22, (C22xC4).325C23, (C2xSD16).50C22, C4.4D4.55C22, C22.735(C22xD4), C2.81(D8:C22), Q8:C4.157C22, (C22xQ8).332C22, (C2xM4(2)).106C22, (C2xC4).158(C2xD4), (C2xC8.C22):32C2, (C2xC4oD4).190C22, SmallGroup(128,2009)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16:4D4
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=a-1, dad=a3, bc=cb, bd=db, dcd=c-1 >
Subgroups: 472 in 238 conjugacy classes, 94 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), D8, SD16, Q16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C2.D8, C4xD4, C4xD4, C4xQ8, C4:D4, C4:D4, C22:Q8, C22:Q8, C4.4D4, C4.4D4, C2xM4(2), C2xD8, C2xSD16, C2xQ16, C2xQ16, C8.C22, C22xQ8, C2xC4oD4, C8:6D4, C4xQ16, D4:D4, C22:Q16, Q8.D4, C8:D4, C8.12D4, Q8:5D4, C2xC8.C22, Q16:4D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, 2+ 1+4, D42, D8:C22, Q8oD8, Q16:4D4
Character table of Q16:4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | symplectic lifted from Q8oD8, Schur index 2 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 0 | 2√2 | 0 | 0 | symplectic lifted from Q8oD8, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8:C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8:C22 |
►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 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(25 57 29 61)(26 64 30 60)(27 63 31 59)(28 62 32 58)(33 52 37 56)(34 51 38 55)(35 50 39 54)(36 49 40 53)
(1 62 44 40)(2 61 45 39)(3 60 46 38)(4 59 47 37)(5 58 48 36)(6 57 41 35)(7 64 42 34)(8 63 43 33)(9 32 23 53)(10 31 24 52)(11 30 17 51)(12 29 18 50)(13 28 19 49)(14 27 20 56)(15 26 21 55)(16 25 22 54)
(1 50)(2 53)(3 56)(4 51)(5 54)(6 49)(7 52)(8 55)(9 39)(10 34)(11 37)(12 40)(13 35)(14 38)(15 33)(16 36)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 48)(26 43)(27 46)(28 41)(29 44)(30 47)(31 42)(32 45)
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), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,46,21,42)(18,45,22,41)(19,44,23,48)(20,43,24,47)(25,57,29,61)(26,64,30,60)(27,63,31,59)(28,62,32,58)(33,52,37,56)(34,51,38,55)(35,50,39,54)(36,49,40,53), (1,62,44,40)(2,61,45,39)(3,60,46,38)(4,59,47,37)(5,58,48,36)(6,57,41,35)(7,64,42,34)(8,63,43,33)(9,32,23,53)(10,31,24,52)(11,30,17,51)(12,29,18,50)(13,28,19,49)(14,27,20,56)(15,26,21,55)(16,25,22,54), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,39)(10,34)(11,37)(12,40)(13,35)(14,38)(15,33)(16,36)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45)>;
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), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,46,21,42)(18,45,22,41)(19,44,23,48)(20,43,24,47)(25,57,29,61)(26,64,30,60)(27,63,31,59)(28,62,32,58)(33,52,37,56)(34,51,38,55)(35,50,39,54)(36,49,40,53), (1,62,44,40)(2,61,45,39)(3,60,46,38)(4,59,47,37)(5,58,48,36)(6,57,41,35)(7,64,42,34)(8,63,43,33)(9,32,23,53)(10,31,24,52)(11,30,17,51)(12,29,18,50)(13,28,19,49)(14,27,20,56)(15,26,21,55)(16,25,22,54), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,39)(10,34)(11,37)(12,40)(13,35)(14,38)(15,33)(16,36)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45) );
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)], [(1,9,5,13),(2,16,6,12),(3,15,7,11),(4,14,8,10),(17,46,21,42),(18,45,22,41),(19,44,23,48),(20,43,24,47),(25,57,29,61),(26,64,30,60),(27,63,31,59),(28,62,32,58),(33,52,37,56),(34,51,38,55),(35,50,39,54),(36,49,40,53)], [(1,62,44,40),(2,61,45,39),(3,60,46,38),(4,59,47,37),(5,58,48,36),(6,57,41,35),(7,64,42,34),(8,63,43,33),(9,32,23,53),(10,31,24,52),(11,30,17,51),(12,29,18,50),(13,28,19,49),(14,27,20,56),(15,26,21,55),(16,25,22,54)], [(1,50),(2,53),(3,56),(4,51),(5,54),(6,49),(7,52),(8,55),(9,39),(10,34),(11,37),(12,40),(13,35),(14,38),(15,33),(16,36),(17,59),(18,62),(19,57),(20,60),(21,63),(22,58),(23,61),(24,64),(25,48),(26,43),(27,46),(28,41),(29,44),(30,47),(31,42),(32,45)]])
Matrix representation of Q16:4D4 ►in GL6(F17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 7 | 0 |
| 0 | 0 | 14 | 3 | 0 | 10 |
| 0 | 0 | 0 | 7 | 14 | 3 |
| 0 | 0 | 7 | 0 | 14 | 14 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 1 |
| 0 | 0 | 4 | 4 | 16 | 0 |
| 0 | 0 | 0 | 16 | 13 | 4 |
| 0 | 0 | 1 | 0 | 4 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 3 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,14,0,7,0,0,3,3,7,0,0,0,7,0,14,14,0,0,0,10,3,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,4,0,1,0,0,4,4,16,0,0,0,0,16,13,4,0,0,1,0,4,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;
Q16:4D4 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_4D_4 % in TeX
G:=Group("Q16:4D4"); // GroupNames label
G:=SmallGroup(128,2009);
// by ID
G=gap.SmallGroup(128,2009);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,346,248,2804,1411,375,172]);
// 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=a^-1,d*a*d=a^3,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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