p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊3Q8, C42.69C23, C4.1032- 1+4, C8⋊Q8⋊34C2, C8.8(C2×Q8), C2.46(D4×Q8), C4⋊C4.391D4, C8⋊4Q8⋊17C2, Q8.Q8⋊52C2, D4.13(C2×Q8), Q8.13(C2×Q8), D4.Q8.4C2, Q8⋊Q8⋊29C2, Q8⋊3Q8⋊11C2, D4⋊2Q8.3C2, (C2×Q8).140D4, C8.5Q8⋊24C2, (C4×SD16).7C2, C4.46(C22×Q8), C4⋊C4.277C23, C4⋊C8.147C22, (C2×C4).580C24, (C2×C8).377C23, (C4×C8).204C22, D4⋊3Q8.11C2, C4⋊Q8.209C22, C8⋊C4.73C22, SD16⋊C4.5C2, (C4×D4).215C22, (C2×D4).440C23, (C4×Q8).207C22, (C2×Q8).414C23, C2.D8.141C22, C4.Q8.120C22, C2.110(D4○SD16), D4⋊C4.96C22, C22.840(C22×D4), C42.C2.78C22, Q8⋊C4.188C22, (C2×SD16).125C22, C2.105(D8⋊C22), (C2×C4).650(C2×D4), SmallGroup(128,2120)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊3Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, cbc-1=dbd-1=a4b, dcd-1=c-1 >
Subgroups: 288 in 167 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C2×SD16, C4×SD16, SD16⋊C4, C8⋊4Q8, Q8⋊Q8, D4⋊2Q8, D4.Q8, Q8.Q8, C8.5Q8, C8⋊Q8, D4⋊3Q8, Q8⋊3Q8, SD16⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8, D8⋊C22, D4○SD16, SD16⋊3Q8
Character table of SD16⋊3Q8
| 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 | 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 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 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 | 4i | 0 | -4i | 0 | 0 | 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 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| 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 | complex lifted from D4○SD16 |
►On 64 points
Generators in S64
(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 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 52)(10 55)(11 50)(12 53)(13 56)(14 51)(15 54)(16 49)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)(33 61)(34 64)(35 59)(36 62)(37 57)(38 60)(39 63)(40 58)
(1 16 23 53)(2 9 24 54)(3 10 17 55)(4 11 18 56)(5 12 19 49)(6 13 20 50)(7 14 21 51)(8 15 22 52)(25 64 41 36)(26 57 42 37)(27 58 43 38)(28 59 44 39)(29 60 45 40)(30 61 46 33)(31 62 47 34)(32 63 48 35)
(1 26 23 42)(2 31 24 47)(3 28 17 44)(4 25 18 41)(5 30 19 46)(6 27 20 43)(7 32 21 48)(8 29 22 45)(9 34 54 62)(10 39 55 59)(11 36 56 64)(12 33 49 61)(13 38 50 58)(14 35 51 63)(15 40 52 60)(16 37 53 57)
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), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,52)(10,55)(11,50)(12,53)(13,56)(14,51)(15,54)(16,49)(25,43)(26,46)(27,41)(28,44)(29,47)(30,42)(31,45)(32,48)(33,61)(34,64)(35,59)(36,62)(37,57)(38,60)(39,63)(40,58), (1,16,23,53)(2,9,24,54)(3,10,17,55)(4,11,18,56)(5,12,19,49)(6,13,20,50)(7,14,21,51)(8,15,22,52)(25,64,41,36)(26,57,42,37)(27,58,43,38)(28,59,44,39)(29,60,45,40)(30,61,46,33)(31,62,47,34)(32,63,48,35), (1,26,23,42)(2,31,24,47)(3,28,17,44)(4,25,18,41)(5,30,19,46)(6,27,20,43)(7,32,21,48)(8,29,22,45)(9,34,54,62)(10,39,55,59)(11,36,56,64)(12,33,49,61)(13,38,50,58)(14,35,51,63)(15,40,52,60)(16,37,53,57)>;
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), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,52)(10,55)(11,50)(12,53)(13,56)(14,51)(15,54)(16,49)(25,43)(26,46)(27,41)(28,44)(29,47)(30,42)(31,45)(32,48)(33,61)(34,64)(35,59)(36,62)(37,57)(38,60)(39,63)(40,58), (1,16,23,53)(2,9,24,54)(3,10,17,55)(4,11,18,56)(5,12,19,49)(6,13,20,50)(7,14,21,51)(8,15,22,52)(25,64,41,36)(26,57,42,37)(27,58,43,38)(28,59,44,39)(29,60,45,40)(30,61,46,33)(31,62,47,34)(32,63,48,35), (1,26,23,42)(2,31,24,47)(3,28,17,44)(4,25,18,41)(5,30,19,46)(6,27,20,43)(7,32,21,48)(8,29,22,45)(9,34,54,62)(10,39,55,59)(11,36,56,64)(12,33,49,61)(13,38,50,58)(14,35,51,63)(15,40,52,60)(16,37,53,57) );
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)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,52),(10,55),(11,50),(12,53),(13,56),(14,51),(15,54),(16,49),(25,43),(26,46),(27,41),(28,44),(29,47),(30,42),(31,45),(32,48),(33,61),(34,64),(35,59),(36,62),(37,57),(38,60),(39,63),(40,58)], [(1,16,23,53),(2,9,24,54),(3,10,17,55),(4,11,18,56),(5,12,19,49),(6,13,20,50),(7,14,21,51),(8,15,22,52),(25,64,41,36),(26,57,42,37),(27,58,43,38),(28,59,44,39),(29,60,45,40),(30,61,46,33),(31,62,47,34),(32,63,48,35)], [(1,26,23,42),(2,31,24,47),(3,28,17,44),(4,25,18,41),(5,30,19,46),(6,27,20,43),(7,32,21,48),(8,29,22,45),(9,34,54,62),(10,39,55,59),(11,36,56,64),(12,33,49,61),(13,38,50,58),(14,35,51,63),(15,40,52,60),(16,37,53,57)]])
Matrix representation of SD16⋊3Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 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 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,5,5,0,0,0,0,12,5,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0] >;
SD16⋊3Q8 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_3Q_8 % in TeX
G:=Group("SD16:3Q8"); // GroupNames label
G:=SmallGroup(128,2120);
// by ID
G=gap.SmallGroup(128,2120);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,346,304,4037,1027,124]);
// 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,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations
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