p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊5D4, C42.445C23, C4.1322+ 1+4, C2.62D42, (D4×Q8)⋊5C2, C8⋊6D4⋊9C2, C8⋊5D4⋊9C2, C8.10(C2×D4), C8⋊D4⋊33C2, C4⋊C4.362D4, (C4×Q16)⋊32C2, Q8.25(C2×D4), C4⋊SD16⋊20C2, Q8⋊D4⋊19C2, C4⋊2Q16⋊37C2, (C2×D4).312D4, C4⋊C8.97C22, C4⋊3(C8.C22), Q8⋊6D4.4C2, C22⋊C4.45D4, C4.92(C22×D4), D4.7D4⋊41C2, C4⋊C4.217C23, (C2×C4).476C24, (C2×C8).286C23, (C4×C8).191C22, C23.318(C2×D4), C4⋊Q8.136C22, C2.62(D4○SD16), (C2×D4).214C23, (C4×D4).150C22, C4⋊D4.64C22, C4⋊1D4.77C22, C22⋊C8.75C22, (C4×Q8).142C22, (C2×Q8).198C23, C2.D8.223C22, C22⋊Q8.63C22, D4⋊C4.68C22, (C22×C4).326C23, (C2×Q16).161C22, (C2×SD16).93C22, C22.736(C22×D4), Q8⋊C4.177C22, (C22×Q8).333C22, (C2×M4(2)).107C22, (C2×C4).159(C2×D4), (C2×C8.C22)⋊33C2, C2.73(C2×C8.C22), (C2×C4○D4).191C22, SmallGroup(128,2010)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊5D4
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >
Subgroups: 488 in 241 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4⋊1D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C8⋊6D4, C4×Q16, Q8⋊D4, D4.7D4, C4⋊SD16, C4⋊2Q16, C8⋊D4, C8⋊5D4, D4×Q8, Q8⋊6D4, C2×C8.C22, Q16⋊5D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2+ 1+4, D42, C2×C8.C22, D4○SD16, Q16⋊5D4
Character table of Q16⋊5D4
| 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 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -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 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 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 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 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 | 4 | 0 | 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 | -4 | 0 | 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 | 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
(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 11 5 15)(2 10 6 14)(3 9 7 13)(4 16 8 12)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 36 19 60)(2 37 20 61)(3 38 21 62)(4 39 22 63)(5 40 23 64)(6 33 24 57)(7 34 17 58)(8 35 18 59)(9 31 53 45)(10 32 54 46)(11 25 55 47)(12 26 56 48)(13 27 49 41)(14 28 50 42)(15 29 51 43)(16 30 52 44)
(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(25 63)(26 60)(27 57)(28 62)(29 59)(30 64)(31 61)(32 58)(33 41)(34 46)(35 43)(36 48)(37 45)(38 42)(39 47)(40 44)
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), (1,11,5,15)(2,10,6,14)(3,9,7,13)(4,16,8,12)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63), (1,36,19,60)(2,37,20,61)(3,38,21,62)(4,39,22,63)(5,40,23,64)(6,33,24,57)(7,34,17,58)(8,35,18,59)(9,31,53,45)(10,32,54,46)(11,25,55,47)(12,26,56,48)(13,27,49,41)(14,28,50,42)(15,29,51,43)(16,30,52,44), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(33,41)(34,46)(35,43)(36,48)(37,45)(38,42)(39,47)(40,44)>;
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), (1,11,5,15)(2,10,6,14)(3,9,7,13)(4,16,8,12)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63), (1,36,19,60)(2,37,20,61)(3,38,21,62)(4,39,22,63)(5,40,23,64)(6,33,24,57)(7,34,17,58)(8,35,18,59)(9,31,53,45)(10,32,54,46)(11,25,55,47)(12,26,56,48)(13,27,49,41)(14,28,50,42)(15,29,51,43)(16,30,52,44), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(33,41)(34,46)(35,43)(36,48)(37,45)(38,42)(39,47)(40,44) );
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)], [(1,11,5,15),(2,10,6,14),(3,9,7,13),(4,16,8,12),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(25,40,29,36),(26,39,30,35),(27,38,31,34),(28,37,32,33),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63)], [(1,36,19,60),(2,37,20,61),(3,38,21,62),(4,39,22,63),(5,40,23,64),(6,33,24,57),(7,34,17,58),(8,35,18,59),(9,31,53,45),(10,32,54,46),(11,25,55,47),(12,26,56,48),(13,27,49,41),(14,28,50,42),(15,29,51,43),(16,30,52,44)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15),(17,54),(18,51),(19,56),(20,53),(21,50),(22,55),(23,52),(24,49),(25,63),(26,60),(27,57),(28,62),(29,59),(30,64),(31,61),(32,58),(33,41),(34,46),(35,43),(36,48),(37,45),(38,42),(39,47),(40,44)]])
Matrix representation of Q16⋊5D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,12,5,0,0,0,0,5,5,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1,0,0,0],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16,0,0,0] >;
Q16⋊5D4 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_5D_4 % in TeX
G:=Group("Q16:5D4"); // GroupNames label
G:=SmallGroup(128,2010);
// by ID
G=gap.SmallGroup(128,2010);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,346,248,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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