p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16:10D4, C42.38C23, C4.1252+ 1+4, C2.55D42, C8:9D4:8C2, C8.34(C2xD4), C8:2D4:20C2, C8:8D4:20C2, C8:3D4:19C2, C4:C4.358D4, Q8:6D4:3C2, Q8:5D4:3C2, Q8.21(C2xD4), D4:D4:36C2, Q8:D4:17C2, C4:SD16:18C2, (C2xD4).163D4, C4:C8.94C22, (C2xC8).88C23, C4.85(C22xD4), Q16:C4:20C2, D4.7D4:37C2, C4:C4.210C23, (C2xC4).469C24, Q8.D4:37C2, C22:C4.159D4, (C2xD8).80C22, C23.102(C2xD4), C4.Q8.53C22, C8:C4.38C22, C2.59(D4oSD16), (C2xD4).209C23, (C4xD4).145C22, C4:1D4.75C22, C4:D4.60C22, C22:C8.72C22, (C4xQ8).139C22, (C2xQ8).388C23, C22:Q8.59C22, D4:C4.66C22, (C22xC8).285C22, (C2xQ16).160C22, Q8:C4.67C22, (C2xSD16).49C22, C4.4D4.54C22, C22.729(C22xD4), C2.78(D8:C22), (C22xC4).1120C23, (C22xQ8).330C22, (C2xM4(2)).104C22, (C2xC4oD8):26C2, (C2xC4).593(C2xD4), (C2xC8.C22):29C2, (C2xC4oD4).186C22, SmallGroup(128,2003)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16:10D4
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=a-1, cac-1=dad=a3, cbc-1=a4b, bd=db, dcd=c-1 >
Subgroups: 512 in 244 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), D8, SD16, Q16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C8:C4, C22:C8, D4:C4, Q8:C4, C4:C8, C4.Q8, C4xD4, C4xD4, C4xQ8, C4:D4, C4:D4, C22:Q8, C22:Q8, C4.4D4, C4.4D4, C4:1D4, C4:1D4, C22xC8, C2xM4(2), C2xD8, C2xSD16, C2xQ16, C4oD8, C8.C22, C22xQ8, C2xC4oD4, C8:9D4, Q16:C4, Q8:D4, D4:D4, D4.7D4, C4:SD16, Q8.D4, C8:8D4, C8:2D4, C8:3D4, Q8:5D4, Q8:6D4, C2xC4oD8, C2xC8.C22, Q16:10D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, 2+ 1+4, D42, D8:C22, D4oSD16, Q16:10D4
Character table of Q16:10D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 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 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8:C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8:C22 |
| ρ28 | 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 | complex lifted from D4oSD16 |
| ρ29 | 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 | complex lifted from D4oSD16 |
►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 11 5 15)(2 10 6 14)(3 9 7 13)(4 16 8 12)(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 56 37 52)(34 55 38 51)(35 54 39 50)(36 53 40 49)
(1 64 48 34)(2 59 41 37)(3 62 42 40)(4 57 43 35)(5 60 44 38)(6 63 45 33)(7 58 46 36)(8 61 47 39)(9 28 17 53)(10 31 18 56)(11 26 19 51)(12 29 20 54)(13 32 21 49)(14 27 22 52)(15 30 23 55)(16 25 24 50)
(1 50)(2 53)(3 56)(4 51)(5 54)(6 49)(7 52)(8 55)(9 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(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), (1,11,5,15)(2,10,6,14)(3,9,7,13)(4,16,8,12)(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)(33,56,37,52)(34,55,38,51)(35,54,39,50)(36,53,40,49), (1,64,48,34)(2,59,41,37)(3,62,42,40)(4,57,43,35)(5,60,44,38)(6,63,45,33)(7,58,46,36)(8,61,47,39)(9,28,17,53)(10,31,18,56)(11,26,19,51)(12,29,20,54)(13,32,21,49)(14,27,22,52)(15,30,23,55)(16,25,24,50), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(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), (1,11,5,15)(2,10,6,14)(3,9,7,13)(4,16,8,12)(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)(33,56,37,52)(34,55,38,51)(35,54,39,50)(36,53,40,49), (1,64,48,34)(2,59,41,37)(3,62,42,40)(4,57,43,35)(5,60,44,38)(6,63,45,33)(7,58,46,36)(8,61,47,39)(9,28,17,53)(10,31,18,56)(11,26,19,51)(12,29,20,54)(13,32,21,49)(14,27,22,52)(15,30,23,55)(16,25,24,50), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(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)], [(1,11,5,15),(2,10,6,14),(3,9,7,13),(4,16,8,12),(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),(33,56,37,52),(34,55,38,51),(35,54,39,50),(36,53,40,49)], [(1,64,48,34),(2,59,41,37),(3,62,42,40),(4,57,43,35),(5,60,44,38),(6,63,45,33),(7,58,46,36),(8,61,47,39),(9,28,17,53),(10,31,18,56),(11,26,19,51),(12,29,20,54),(13,32,21,49),(14,27,22,52),(15,30,23,55),(16,25,24,50)], [(1,50),(2,53),(3,56),(4,51),(5,54),(6,49),(7,52),(8,55),(9,37),(10,40),(11,35),(12,38),(13,33),(14,36),(15,39),(16,34),(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 Q16:10D4 ►in GL6(F17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 4 | 0 | 0 | 13 |
| 0 | 0 | 0 | 4 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 1 | 0 | 16 | 0 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 12 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 3 | 0 |
| 0 | 0 | 14 | 6 | 10 | 7 |
| 0 | 0 | 11 | 3 | 13 | 3 |
| 0 | 0 | 14 | 10 | 13 | 14 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 5 |
| 0 | 0 | 1 | 0 | 11 | 11 |
| 0 | 0 | 6 | 6 | 12 | 16 |
| 0 | 0 | 12 | 12 | 5 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,9,13,13,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,16,0,0,15,16,16,16,0,0,0,1,0,0],[3,12,0,0,0,0,2,14,0,0,0,0,0,0,1,14,11,14,0,0,6,6,3,10,0,0,3,10,13,13,0,0,0,7,3,14],[14,4,0,0,0,0,15,3,0,0,0,0,0,0,6,1,6,12,0,0,7,0,6,12,0,0,0,11,12,5,0,0,5,11,16,16] >;
Q16:10D4 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_{10}D_4 % in TeX
G:=Group("Q16:10D4"); // GroupNames label
G:=SmallGroup(128,2003);
// by ID
G=gap.SmallGroup(128,2003);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,346,248,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^3,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export