p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊3D4, C42.443C23, C4.1302+ 1+4, C2.60D42, (D4×Q8)⋊4C2, C8.8(C2×D4), C8⋊6D4⋊7C2, C4⋊C4.361D4, D4.25(C2×D4), Q8.23(C2×D4), C4⋊2Q16⋊36C2, C4⋊Q16⋊22C2, (C4×SD16)⋊21C2, D4⋊6D4.4C2, (C2×D4).311D4, C8.D4⋊22C2, C4⋊C8.95C22, C4⋊2(C8.C22), C2.39(Q8○D8), C22⋊C4.44D4, C4.90(C22×D4), D4.D4⋊20C2, D4.7D4⋊40C2, C4⋊C4.215C23, (C2×C8).285C23, (C4×C8).189C22, (C2×C4).474C24, C22⋊Q16⋊28C2, C23.316(C2×D4), C4⋊Q8.135C22, (C2×D4).415C23, (C4×D4).148C22, C22⋊C8.73C22, (C4×Q8).140C22, (C2×Q8).197C23, (C2×Q16).33C22, C4.Q8.136C22, C22⋊Q8.61C22, (C22×C4).324C23, C22.734(C22×D4), D4⋊C4.165C22, Q8⋊C4.156C22, (C2×SD16).119C22, (C22×Q8).331C22, (C2×M4(2)).105C22, (C2×C4).594(C2×D4), (C2×C8.C22)⋊31C2, C2.72(C2×C8.C22), (C2×C4○D4).189C22, SmallGroup(128,2008)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊3D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 448 in 236 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, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4⋊Q8, C4⋊Q8, C2×M4(2), C2×SD16, C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C8⋊6D4, C4×SD16, C22⋊Q16, D4.7D4, D4.D4, C4⋊2Q16, C8.D4, C4⋊Q16, D4⋊6D4, D4×Q8, C2×C8.C22, SD16⋊3D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2+ 1+4, D42, C2×C8.C22, Q8○D8, SD16⋊3D4
Character table of SD16⋊3D4
| 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 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 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 | 4 | 0 | -4 | 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 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 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 | 2√2 | -2√2 | 0 | 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 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)(17 43)(18 46)(19 41)(20 44)(21 47)(22 42)(23 45)(24 48)(25 35)(26 38)(27 33)(28 36)(29 39)(30 34)(31 37)(32 40)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)
(1 24 14 48)(2 17 15 41)(3 18 16 42)(4 19 9 43)(5 20 10 44)(6 21 11 45)(7 22 12 46)(8 23 13 47)(25 52 39 63)(26 53 40 64)(27 54 33 57)(28 55 34 58)(29 56 35 59)(30 49 36 60)(31 50 37 61)(32 51 38 62)
(1 56)(2 53)(3 50)(4 55)(5 52)(6 49)(7 54)(8 51)(9 58)(10 63)(11 60)(12 57)(13 62)(14 59)(15 64)(16 61)(17 26)(18 31)(19 28)(20 25)(21 30)(22 27)(23 32)(24 29)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)
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), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11)(17,43)(18,46)(19,41)(20,44)(21,47)(22,42)(23,45)(24,48)(25,35)(26,38)(27,33)(28,36)(29,39)(30,34)(31,37)(32,40)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63), (1,24,14,48)(2,17,15,41)(3,18,16,42)(4,19,9,43)(5,20,10,44)(6,21,11,45)(7,22,12,46)(8,23,13,47)(25,52,39,63)(26,53,40,64)(27,54,33,57)(28,55,34,58)(29,56,35,59)(30,49,36,60)(31,50,37,61)(32,51,38,62), (1,56)(2,53)(3,50)(4,55)(5,52)(6,49)(7,54)(8,51)(9,58)(10,63)(11,60)(12,57)(13,62)(14,59)(15,64)(16,61)(17,26)(18,31)(19,28)(20,25)(21,30)(22,27)(23,32)(24,29)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)>;
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), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11)(17,43)(18,46)(19,41)(20,44)(21,47)(22,42)(23,45)(24,48)(25,35)(26,38)(27,33)(28,36)(29,39)(30,34)(31,37)(32,40)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63), (1,24,14,48)(2,17,15,41)(3,18,16,42)(4,19,9,43)(5,20,10,44)(6,21,11,45)(7,22,12,46)(8,23,13,47)(25,52,39,63)(26,53,40,64)(27,54,33,57)(28,55,34,58)(29,56,35,59)(30,49,36,60)(31,50,37,61)(32,51,38,62), (1,56)(2,53)(3,50)(4,55)(5,52)(6,49)(7,54)(8,51)(9,58)(10,63)(11,60)(12,57)(13,62)(14,59)(15,64)(16,61)(17,26)(18,31)(19,28)(20,25)(21,30)(22,27)(23,32)(24,29)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41) );
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)], [(1,14),(2,9),(3,12),(4,15),(5,10),(6,13),(7,16),(8,11),(17,43),(18,46),(19,41),(20,44),(21,47),(22,42),(23,45),(24,48),(25,35),(26,38),(27,33),(28,36),(29,39),(30,34),(31,37),(32,40),(49,58),(50,61),(51,64),(52,59),(53,62),(54,57),(55,60),(56,63)], [(1,24,14,48),(2,17,15,41),(3,18,16,42),(4,19,9,43),(5,20,10,44),(6,21,11,45),(7,22,12,46),(8,23,13,47),(25,52,39,63),(26,53,40,64),(27,54,33,57),(28,55,34,58),(29,56,35,59),(30,49,36,60),(31,50,37,61),(32,51,38,62)], [(1,56),(2,53),(3,50),(4,55),(5,52),(6,49),(7,54),(8,51),(9,58),(10,63),(11,60),(12,57),(13,62),(14,59),(15,64),(16,61),(17,26),(18,31),(19,28),(20,25),(21,30),(22,27),(23,32),(24,29),(33,46),(34,43),(35,48),(36,45),(37,42),(38,47),(39,44),(40,41)]])
Matrix representation of SD16⋊3D4 ►in GL8(𝔽17)
| 0 | 0 | 16 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 14 | 14 |
| 0 | 0 | 0 | 0 | 4 | 13 | 3 | 14 |
| 0 | 0 | 0 | 0 | 14 | 14 | 4 | 4 |
| 0 | 0 | 0 | 0 | 3 | 14 | 13 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,4,14,3,0,0,0,0,13,13,14,14,0,0,0,0,14,3,4,13,0,0,0,0,14,14,4,4],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0] >;
SD16⋊3D4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_3D_4 % in TeX
G:=Group("SD16:3D4"); // GroupNames label
G:=SmallGroup(128,2008);
// by ID
G=gap.SmallGroup(128,2008);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,352,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations
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