p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16:1Q8, C42.66C23, C4.1002- 1+4, Q82:7C2, C8:Q8:32C2, C8.6(C2xQ8), C2.43(D4xQ8), C4:C4.388D4, C8:4Q8:15C2, C8:2Q8:33C2, Q8.Q8:51C2, Q8.11(C2xQ8), D4.11(C2xQ8), C4.Q16:46C2, C2.68(D4oD8), (C2xQ8).249D4, D4:2Q8.2C2, (C4xSD16).5C2, C4.43(C22xQ8), C4:C8.144C22, C4:C4.274C23, (C2xC4).577C24, (C2xC8).214C23, (C4xC8).201C22, D4:3Q8.10C2, D4:Q8.12C2, C4:Q8.206C22, C2.D8.75C22, C8:C4.70C22, C4.Q8.77C22, SD16:C4.3C2, (C2xD4).438C23, (C4xD4).213C22, C4.53(C8.C22), (C2xQ8).411C23, (C4xQ8).204C22, D4:C4.94C22, C22.837(C22xD4), C42.C2.75C22, Q8:C4.165C22, (C2xSD16).123C22, (C2xC4).647(C2xD4), C2.90(C2xC8.C22), SmallGroup(128,2117)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16:Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 304 in 170 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C42, C42, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C2xQ8, C4xC8, C8:C4, D4:C4, D4:C4, Q8:C4, Q8:C4, C4:C8, C4:C8, C4.Q8, C4.Q8, C2.D8, C2xC4:C4, C4xD4, C4xD4, C4xQ8, C4xQ8, C4xQ8, C22:Q8, C42.C2, C4:Q8, C4:Q8, C4:Q8, C2xSD16, C4xSD16, SD16:C4, C8:4Q8, D4:Q8, D4:2Q8, C4.Q16, Q8.Q8, C8:2Q8, C8:Q8, D4:3Q8, Q82, SD16:Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2xD4, C2xQ8, C24, C8.C22, C22xD4, C22xQ8, 2- 1+4, D4xQ8, C2xC8.C22, D4oD8, SD16:Q8
Character table of SD16:Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | -2 | 2 | -2 | 2 | -2 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | -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 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | -2 | 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 | -2 | 2 | -2 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | orthogonal lifted from D4oD8 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | orthogonal lifted from D4oD8 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ28 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►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 19)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 57)(10 60)(11 63)(12 58)(13 61)(14 64)(15 59)(16 62)(25 53)(26 56)(27 51)(28 54)(29 49)(30 52)(31 55)(32 50)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)
(1 31 23 51)(2 32 24 52)(3 25 17 53)(4 26 18 54)(5 27 19 55)(6 28 20 56)(7 29 21 49)(8 30 22 50)(9 36 61 48)(10 37 62 41)(11 38 63 42)(12 39 64 43)(13 40 57 44)(14 33 58 45)(15 34 59 46)(16 35 60 47)
(1 38 23 42)(2 35 24 47)(3 40 17 44)(4 37 18 41)(5 34 19 46)(6 39 20 43)(7 36 21 48)(8 33 22 45)(9 49 61 29)(10 54 62 26)(11 51 63 31)(12 56 64 28)(13 53 57 25)(14 50 58 30)(15 55 59 27)(16 52 60 32)
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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,57)(10,60)(11,63)(12,58)(13,61)(14,64)(15,59)(16,62)(25,53)(26,56)(27,51)(28,54)(29,49)(30,52)(31,55)(32,50)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48), (1,31,23,51)(2,32,24,52)(3,25,17,53)(4,26,18,54)(5,27,19,55)(6,28,20,56)(7,29,21,49)(8,30,22,50)(9,36,61,48)(10,37,62,41)(11,38,63,42)(12,39,64,43)(13,40,57,44)(14,33,58,45)(15,34,59,46)(16,35,60,47), (1,38,23,42)(2,35,24,47)(3,40,17,44)(4,37,18,41)(5,34,19,46)(6,39,20,43)(7,36,21,48)(8,33,22,45)(9,49,61,29)(10,54,62,26)(11,51,63,31)(12,56,64,28)(13,53,57,25)(14,50,58,30)(15,55,59,27)(16,52,60,32)>;
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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,57)(10,60)(11,63)(12,58)(13,61)(14,64)(15,59)(16,62)(25,53)(26,56)(27,51)(28,54)(29,49)(30,52)(31,55)(32,50)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48), (1,31,23,51)(2,32,24,52)(3,25,17,53)(4,26,18,54)(5,27,19,55)(6,28,20,56)(7,29,21,49)(8,30,22,50)(9,36,61,48)(10,37,62,41)(11,38,63,42)(12,39,64,43)(13,40,57,44)(14,33,58,45)(15,34,59,46)(16,35,60,47), (1,38,23,42)(2,35,24,47)(3,40,17,44)(4,37,18,41)(5,34,19,46)(6,39,20,43)(7,36,21,48)(8,33,22,45)(9,49,61,29)(10,54,62,26)(11,51,63,31)(12,56,64,28)(13,53,57,25)(14,50,58,30)(15,55,59,27)(16,52,60,32) );
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)], [(1,19),(2,22),(3,17),(4,20),(5,23),(6,18),(7,21),(8,24),(9,57),(10,60),(11,63),(12,58),(13,61),(14,64),(15,59),(16,62),(25,53),(26,56),(27,51),(28,54),(29,49),(30,52),(31,55),(32,50),(33,43),(34,46),(35,41),(36,44),(37,47),(38,42),(39,45),(40,48)], [(1,31,23,51),(2,32,24,52),(3,25,17,53),(4,26,18,54),(5,27,19,55),(6,28,20,56),(7,29,21,49),(8,30,22,50),(9,36,61,48),(10,37,62,41),(11,38,63,42),(12,39,64,43),(13,40,57,44),(14,33,58,45),(15,34,59,46),(16,35,60,47)], [(1,38,23,42),(2,35,24,47),(3,40,17,44),(4,37,18,41),(5,34,19,46),(6,39,20,43),(7,36,21,48),(8,33,22,45),(9,49,61,29),(10,54,62,26),(11,51,63,31),(12,56,64,28),(13,53,57,25),(14,50,58,30),(15,55,59,27),(16,52,60,32)]])
Matrix representation of SD16:Q8 ►in GL6(F17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 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 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
SD16:Q8 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes Q_8 % in TeX
G:=Group("SD16:Q8"); // GroupNames label
G:=SmallGroup(128,2117);
// by ID
G=gap.SmallGroup(128,2117);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations
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