p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊8D4, C42.36C23, C4.1232+ 1+4, C2.53D42, C8⋊9D4⋊6C2, C8.32(C2×D4), D4⋊6D4⋊3C2, C8⋊D4⋊29C2, C4⋊C4.357D4, Q8⋊5D4⋊2C2, D4.21(C2×D4), Q8.19(C2×D4), D4⋊D4⋊35C2, C4⋊2Q16⋊34C2, (C2×D4).162D4, C8.2D4⋊18C2, C4⋊C8.92C22, (C2×C8).86C23, C2.37(Q8○D8), C4.83(C22×D4), D4.7D4⋊35C2, D4.2D4⋊37C2, C8.18D4⋊30C2, C4⋊C4.208C23, (C2×C4).467C24, C22⋊Q16⋊26C2, C22⋊C4.158D4, C23.101(C2×D4), C4⋊Q8.133C22, SD16⋊C4⋊30C2, C8⋊C4.36C22, (C2×D4).207C23, (C2×D8).135C22, (C4×D4).143C22, C4⋊D4.58C22, C22⋊C8.70C22, (C4×Q8).137C22, (C2×Q8).193C23, (C2×Q16).80C22, C2.D8.116C22, C22⋊Q8.57C22, D4⋊C4.65C22, (C22×C8).283C22, Q8⋊C4.65C22, C4.4D4.52C22, C22.727(C22×D4), C2.77(D8⋊C22), (C22×C4).1119C23, (C2×SD16).118C22, (C22×Q8).328C22, (C2×M4(2)).102C22, (C2×C4○D8)⋊25C2, (C2×C4).591(C2×D4), (C2×C8.C22)⋊27C2, (C2×C4○D4).184C22, SmallGroup(128,2001)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊8D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 472 in 239 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, 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), D8, SD16, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C8.C22, C22×Q8, C2×C4○D4, C8⋊9D4, SD16⋊C4, D4⋊D4, C22⋊Q16, D4.7D4, C4⋊2Q16, D4.2D4, C8.18D4, C8⋊D4, C8.2D4, D4⋊6D4, Q8⋊5D4, C2×C4○D8, C2×C8.C22, SD16⋊8D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8⋊C22, Q8○D8, SD16⋊8D4
Character table of SD16⋊8D4
| 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 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | -2 | 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 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ27 | 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 | 0 | -2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►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 12)(2 15)(3 10)(4 13)(5 16)(6 11)(7 14)(8 9)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 64)(26 59)(27 62)(28 57)(29 60)(30 63)(31 58)(32 61)(33 52)(34 55)(35 50)(36 53)(37 56)(38 51)(39 54)(40 49)
(1 27 45 49)(2 32 46 54)(3 29 47 51)(4 26 48 56)(5 31 41 53)(6 28 42 50)(7 25 43 55)(8 30 44 52)(9 63 24 33)(10 60 17 38)(11 57 18 35)(12 62 19 40)(13 59 20 37)(14 64 21 34)(15 61 22 39)(16 58 23 36)
(1 37)(2 34)(3 39)(4 36)(5 33)(6 38)(7 35)(8 40)(9 53)(10 50)(11 55)(12 52)(13 49)(14 54)(15 51)(16 56)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 63)(42 60)(43 57)(44 62)(45 59)(46 64)(47 61)(48 58)
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), (1,12)(2,15)(3,10)(4,13)(5,16)(6,11)(7,14)(8,9)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,64)(26,59)(27,62)(28,57)(29,60)(30,63)(31,58)(32,61)(33,52)(34,55)(35,50)(36,53)(37,56)(38,51)(39,54)(40,49), (1,27,45,49)(2,32,46,54)(3,29,47,51)(4,26,48,56)(5,31,41,53)(6,28,42,50)(7,25,43,55)(8,30,44,52)(9,63,24,33)(10,60,17,38)(11,57,18,35)(12,62,19,40)(13,59,20,37)(14,64,21,34)(15,61,22,39)(16,58,23,36), (1,37)(2,34)(3,39)(4,36)(5,33)(6,38)(7,35)(8,40)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(41,63)(42,60)(43,57)(44,62)(45,59)(46,64)(47,61)(48,58)>;
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), (1,12)(2,15)(3,10)(4,13)(5,16)(6,11)(7,14)(8,9)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,64)(26,59)(27,62)(28,57)(29,60)(30,63)(31,58)(32,61)(33,52)(34,55)(35,50)(36,53)(37,56)(38,51)(39,54)(40,49), (1,27,45,49)(2,32,46,54)(3,29,47,51)(4,26,48,56)(5,31,41,53)(6,28,42,50)(7,25,43,55)(8,30,44,52)(9,63,24,33)(10,60,17,38)(11,57,18,35)(12,62,19,40)(13,59,20,37)(14,64,21,34)(15,61,22,39)(16,58,23,36), (1,37)(2,34)(3,39)(4,36)(5,33)(6,38)(7,35)(8,40)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(41,63)(42,60)(43,57)(44,62)(45,59)(46,64)(47,61)(48,58) );
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)], [(1,12),(2,15),(3,10),(4,13),(5,16),(6,11),(7,14),(8,9),(17,47),(18,42),(19,45),(20,48),(21,43),(22,46),(23,41),(24,44),(25,64),(26,59),(27,62),(28,57),(29,60),(30,63),(31,58),(32,61),(33,52),(34,55),(35,50),(36,53),(37,56),(38,51),(39,54),(40,49)], [(1,27,45,49),(2,32,46,54),(3,29,47,51),(4,26,48,56),(5,31,41,53),(6,28,42,50),(7,25,43,55),(8,30,44,52),(9,63,24,33),(10,60,17,38),(11,57,18,35),(12,62,19,40),(13,59,20,37),(14,64,21,34),(15,61,22,39),(16,58,23,36)], [(1,37),(2,34),(3,39),(4,36),(5,33),(6,38),(7,35),(8,40),(9,53),(10,50),(11,55),(12,52),(13,49),(14,54),(15,51),(16,56),(17,28),(18,25),(19,30),(20,27),(21,32),(22,29),(23,26),(24,31),(41,63),(42,60),(43,57),(44,62),(45,59),(46,64),(47,61),(48,58)]])
Matrix representation of SD16⋊8D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 1 | 0 |
| 0 | 0 | 0 | 10 | 0 | 1 |
| 0 | 0 | 1 | 0 | 7 | 0 |
| 0 | 0 | 0 | 1 | 0 | 7 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 7 |
| 0 | 0 | 16 | 0 | 10 | 0 |
| 0 | 0 | 0 | 7 | 0 | 16 |
| 0 | 0 | 10 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,3,3,0,0,0,0,14,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,14,3],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,10,0,1,0,0,0,0,10,0,1,0,0,1,0,7,0,0,0,0,1,0,7],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,16,0,10,0,0,1,0,7,0,0,0,0,10,0,1,0,0,7,0,16,0] >;
SD16⋊8D4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_8D_4 % in TeX
G:=Group("SD16:8D4"); // GroupNames label
G:=SmallGroup(128,2001);
// by ID
G=gap.SmallGroup(128,2001);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,352,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,c*a*c^-1=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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