p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊2Q8, C42.67C23, C4.1012- 1+4, C8⋊3(C2×Q8), C8⋊Q8⋊33C2, Q8⋊5(C2×Q8), C2.44(D4×Q8), C4⋊C4.389D4, C8⋊2Q8⋊34C2, C8⋊4Q8⋊16C2, D4.12(C2×Q8), (D4×Q8).10C2, D4.Q8.3C2, Q8⋊3Q8⋊10C2, C4.Q16⋊47C2, Q8⋊Q8⋊28C2, (C2×Q8).250D4, C2.68(Q8○D8), (C4×SD16).6C2, C4.44(C22×Q8), C4⋊C8.145C22, C4⋊C4.275C23, C4.80(C8⋊C22), (C2×C8).215C23, (C2×C4).578C24, (C4×C8).202C22, D4⋊Q8.13C2, C4⋊Q8.207C22, C8⋊C4.71C22, C4.Q8.78C22, SD16⋊C4.4C2, (C2×D4).439C23, (C4×D4).214C22, (C2×Q8).412C23, (C4×Q8).205C22, C2.D8.139C22, D4⋊C4.95C22, C22.838(C22×D4), C42.C2.76C22, Q8⋊C4.166C22, (C2×SD16).124C22, (C2×C4).648(C2×D4), C2.91(C2×C8⋊C22), SmallGroup(128,2118)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊2Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 320 in 180 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×SD16, C22×Q8, C4×SD16, SD16⋊C4, C8⋊4Q8, D4⋊Q8, Q8⋊Q8, C4.Q16, D4.Q8, C8⋊2Q8, C8⋊Q8, D4×Q8, Q8⋊3Q8, SD16⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C8⋊C22, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C8⋊C22, Q8○D8, SD16⋊2Q8
Character table of SD16⋊2Q8
| 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 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 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 | orthogonal lifted from C8⋊C22 |
| ρ27 | 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 |
| ρ28 | 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 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ29 | 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 | symplectic lifted from Q8○D8, 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 55)(10 50)(11 53)(12 56)(13 51)(14 54)(15 49)(16 52)(25 62)(26 57)(27 60)(28 63)(29 58)(30 61)(31 64)(32 59)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)
(1 9 23 51)(2 10 24 52)(3 11 17 53)(4 12 18 54)(5 13 19 55)(6 14 20 56)(7 15 21 49)(8 16 22 50)(25 35 60 47)(26 36 61 48)(27 37 62 41)(28 38 63 42)(29 39 64 43)(30 40 57 44)(31 33 58 45)(32 34 59 46)
(1 36 23 48)(2 33 24 45)(3 38 17 42)(4 35 18 47)(5 40 19 44)(6 37 20 41)(7 34 21 46)(8 39 22 43)(9 26 51 61)(10 31 52 58)(11 28 53 63)(12 25 54 60)(13 30 55 57)(14 27 56 62)(15 32 49 59)(16 29 50 64)
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)(9,55)(10,50)(11,53)(12,56)(13,51)(14,54)(15,49)(16,52)(25,62)(26,57)(27,60)(28,63)(29,58)(30,61)(31,64)(32,59)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48), (1,9,23,51)(2,10,24,52)(3,11,17,53)(4,12,18,54)(5,13,19,55)(6,14,20,56)(7,15,21,49)(8,16,22,50)(25,35,60,47)(26,36,61,48)(27,37,62,41)(28,38,63,42)(29,39,64,43)(30,40,57,44)(31,33,58,45)(32,34,59,46), (1,36,23,48)(2,33,24,45)(3,38,17,42)(4,35,18,47)(5,40,19,44)(6,37,20,41)(7,34,21,46)(8,39,22,43)(9,26,51,61)(10,31,52,58)(11,28,53,63)(12,25,54,60)(13,30,55,57)(14,27,56,62)(15,32,49,59)(16,29,50,64)>;
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)(9,55)(10,50)(11,53)(12,56)(13,51)(14,54)(15,49)(16,52)(25,62)(26,57)(27,60)(28,63)(29,58)(30,61)(31,64)(32,59)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48), (1,9,23,51)(2,10,24,52)(3,11,17,53)(4,12,18,54)(5,13,19,55)(6,14,20,56)(7,15,21,49)(8,16,22,50)(25,35,60,47)(26,36,61,48)(27,37,62,41)(28,38,63,42)(29,39,64,43)(30,40,57,44)(31,33,58,45)(32,34,59,46), (1,36,23,48)(2,33,24,45)(3,38,17,42)(4,35,18,47)(5,40,19,44)(6,37,20,41)(7,34,21,46)(8,39,22,43)(9,26,51,61)(10,31,52,58)(11,28,53,63)(12,25,54,60)(13,30,55,57)(14,27,56,62)(15,32,49,59)(16,29,50,64) );
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),(9,55),(10,50),(11,53),(12,56),(13,51),(14,54),(15,49),(16,52),(25,62),(26,57),(27,60),(28,63),(29,58),(30,61),(31,64),(32,59),(33,43),(34,46),(35,41),(36,44),(37,47),(38,42),(39,45),(40,48)], [(1,9,23,51),(2,10,24,52),(3,11,17,53),(4,12,18,54),(5,13,19,55),(6,14,20,56),(7,15,21,49),(8,16,22,50),(25,35,60,47),(26,36,61,48),(27,37,62,41),(28,38,63,42),(29,39,64,43),(30,40,57,44),(31,33,58,45),(32,34,59,46)], [(1,36,23,48),(2,33,24,45),(3,38,17,42),(4,35,18,47),(5,40,19,44),(6,37,20,41),(7,34,21,46),(8,39,22,43),(9,26,51,61),(10,31,52,58),(11,28,53,63),(12,25,54,60),(13,30,55,57),(14,27,56,62),(15,32,49,59),(16,29,50,64)]])
Matrix representation of SD16⋊2Q8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 8 | 7 | 7 |
| 0 | 0 | 9 | 16 | 10 | 7 |
| 0 | 0 | 13 | 4 | 1 | 9 |
| 0 | 0 | 13 | 13 | 8 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 0 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 15 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,9,13,13,0,0,8,16,4,13,0,0,7,10,1,8,0,0,7,7,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,0,0,1,0,0,0,0,0,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,0,1,0,16,0,0,16,0,1,0,0,0,0,2,0,16,0,0,15,0,1,0] >;
SD16⋊2Q8 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_2Q_8 % in TeX
G:=Group("SD16:2Q8"); // GroupNames label
G:=SmallGroup(128,2118);
// by ID
G=gap.SmallGroup(128,2118);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,723,346,80,4037,1027,124]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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