direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5xQ16, C8.9D10, Q8.3D10, Dic20:5C2, D10.25D4, C20.8C23, C40.7C22, Dic5.9D4, Dic10.4C22, C5:2(C2xQ16), (C5xQ16):2C2, C5:Q16:3C2, (C8xD5).1C2, C2.22(D4xD5), (Q8xD5).1C2, C10.34(C2xD4), C4.8(C22xD5), C5:2C8.7C22, (C5xQ8).3C22, (C4xD5).19C22, SmallGroup(160,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5xQ16
G = < a,b,c,d | a5=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 184 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, Q8, Q8, D5, C10, C2xC8, Q16, Q16, C2xQ8, Dic5, Dic5, C20, C20, D10, C2xQ16, C5:2C8, C40, Dic10, Dic10, C4xD5, C4xD5, C5xQ8, C8xD5, Dic20, C5:Q16, C5xQ16, Q8xD5, D5xQ16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2xD4, D10, C2xQ16, C22xD5, D4xD5, D5xQ16
Character table of D5xQ16
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D | |
size | 1 | 1 | 5 | 5 | 2 | 4 | 4 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D10 |
ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
ρ13 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
ρ15 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
ρ16 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
ρ17 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D10 |
ρ18 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | √2 | -√2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | √2 | -√2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -√2 | √2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -√2 | √2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
ρ23 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4xD5 |
ρ24 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4xD5 |
ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2√2 | 2√2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | symplectic faithful, Schur index 2 |
ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2√2 | -2√2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | symplectic faithful, Schur index 2 |
ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2√2 | 2√2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | symplectic faithful, Schur index 2 |
ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2√2 | -2√2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | symplectic faithful, Schur index 2 |
(1 66 62 39 15)(2 67 63 40 16)(3 68 64 33 9)(4 69 57 34 10)(5 70 58 35 11)(6 71 59 36 12)(7 72 60 37 13)(8 65 61 38 14)(17 29 75 50 45)(18 30 76 51 46)(19 31 77 52 47)(20 32 78 53 48)(21 25 79 54 41)(22 26 80 55 42)(23 27 73 56 43)(24 28 74 49 44)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 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)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 75 5 79)(2 74 6 78)(3 73 7 77)(4 80 8 76)(9 27 13 31)(10 26 14 30)(11 25 15 29)(12 32 16 28)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
G:=sub<Sym(80)| (1,66,62,39,15)(2,67,63,40,16)(3,68,64,33,9)(4,69,57,34,10)(5,70,58,35,11)(6,71,59,36,12)(7,72,60,37,13)(8,65,61,38,14)(17,29,75,50,45)(18,30,76,51,46)(19,31,77,52,47)(20,32,78,53,48)(21,25,79,54,41)(22,26,80,55,42)(23,27,73,56,43)(24,28,74,49,44), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,45)(42,46)(43,47)(44,48)(57,61)(58,62)(59,63)(60,64), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75,5,79)(2,74,6,78)(3,73,7,77)(4,80,8,76)(9,27,13,31)(10,26,14,30)(11,25,15,29)(12,32,16,28)(17,35,21,39)(18,34,22,38)(19,33,23,37)(20,40,24,36)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72)>;
G:=Group( (1,66,62,39,15)(2,67,63,40,16)(3,68,64,33,9)(4,69,57,34,10)(5,70,58,35,11)(6,71,59,36,12)(7,72,60,37,13)(8,65,61,38,14)(17,29,75,50,45)(18,30,76,51,46)(19,31,77,52,47)(20,32,78,53,48)(21,25,79,54,41)(22,26,80,55,42)(23,27,73,56,43)(24,28,74,49,44), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,45)(42,46)(43,47)(44,48)(57,61)(58,62)(59,63)(60,64), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75,5,79)(2,74,6,78)(3,73,7,77)(4,80,8,76)(9,27,13,31)(10,26,14,30)(11,25,15,29)(12,32,16,28)(17,35,21,39)(18,34,22,38)(19,33,23,37)(20,40,24,36)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72) );
G=PermutationGroup([[(1,66,62,39,15),(2,67,63,40,16),(3,68,64,33,9),(4,69,57,34,10),(5,70,58,35,11),(6,71,59,36,12),(7,72,60,37,13),(8,65,61,38,14),(17,29,75,50,45),(18,30,76,51,46),(19,31,77,52,47),(20,32,78,53,48),(21,25,79,54,41),(22,26,80,55,42),(23,27,73,56,43),(24,28,74,49,44)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(33,72),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,45),(42,46),(43,47),(44,48),(57,61),(58,62),(59,63),(60,64)], [(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),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,75,5,79),(2,74,6,78),(3,73,7,77),(4,80,8,76),(9,27,13,31),(10,26,14,30),(11,25,15,29),(12,32,16,28),(17,35,21,39),(18,34,22,38),(19,33,23,37),(20,40,24,36),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)]])
D5xQ16 is a maximal subgroup of
D5.Q32 Dic20.C4 SD32:D5 Q32:D5 Dic20:C4 D20.30D4 D20.47D4 D20.44D4 Dic10.D6 D15:Q16
D5xQ16 is a maximal quotient of
Dic5:4Q16 Dic5.3Q16 Dic5:Q16 Dic5.9Q16 D10:4Q16 D10.7Q16 D10:Q16 Dic5:5Q16 C40:2Q8 Dic10:2Q8 D10.8Q16 D10:2Q16 C40.26D4 Dic5:3Q16 D10:5Q16 D10:3Q16 Dic10.D6 D15:Q16
Matrix representation of D5xQ16 ►in GL4(F41) generated by
34 | 1 | 0 | 0 |
40 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 34 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 40 | 0 |
0 | 0 | 0 | 40 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 3 | 0 |
0 | 0 | 0 | 14 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 |
0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,14],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0] >;
D5xQ16 in GAP, Magma, Sage, TeX
D_5\times Q_{16}
% in TeX
G:=Group("D5xQ16");
// GroupNames label
G:=SmallGroup(160,138);
// by ID
G=gap.SmallGroup(160,138);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,116,86,297,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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