metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊2D5, C8.3D10, Q8.4D10, D10.17D4, C20.9C23, C40.10C22, Dic5.19D4, D20.4C22, Dic10.5C22, Q8⋊D5⋊3C2, (Q8×D5)⋊3C2, C8⋊D5⋊4C2, C40⋊C2⋊4C2, (C5×Q16)⋊4C2, C5⋊Q16⋊4C2, C2.23(D4×D5), C10.35(C2×D4), C5⋊3(C8.C22), C4.9(C22×D5), C5⋊2C8.2C22, Q8⋊2D5.1C2, (C4×D5).4C22, (C5×Q8).4C22, SmallGroup(160,139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16⋊D5
G = < a,b,c,d | a8=c5=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 208 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, Q8, D5, C10, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C5×Q8, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, Q16⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8.C22, C22×D5, D4×D5, Q16⋊D5
Character table of Q16⋊D5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 10A | 10B | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 20 | 2 | 4 | 4 | 10 | 20 | 2 | 2 | 4 | 20 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | 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 | -1+√5/2 | -1-√5/2 | 2 | 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 | -1+√5/2 | -1-√5/2 | 2 | 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 | -1+√5/2 | -1-√5/2 | -2 | 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 | -1+√5/2 | -1-√5/2 | -2 | 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 | -1-√5/2 | -1+√5/2 | -2 | 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 | -1-√5/2 | -1+√5/2 | -2 | 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 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 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 | -1-√5/2 | -1+√5/2 | 2 | 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 |
| ρ19 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | -1+√5 | -1-√5 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ20 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | -1-√5 | -1+√5 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 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 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 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 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 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 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 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 | complex faithful |
►On 80 points
Generators in S80
(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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 66 12 41 75)(2 67 13 42 76)(3 68 14 43 77)(4 69 15 44 78)(5 70 16 45 79)(6 71 9 46 80)(7 72 10 47 73)(8 65 11 48 74)(17 29 60 50 40)(18 30 61 51 33)(19 31 62 52 34)(20 32 63 53 35)(21 25 64 54 36)(22 26 57 55 37)(23 27 58 56 38)(24 28 59 49 39)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 37)(10 34)(11 39)(12 36)(13 33)(14 38)(15 35)(16 40)(17 70)(18 67)(19 72)(20 69)(21 66)(22 71)(23 68)(24 65)(41 54)(42 51)(43 56)(44 53)(45 50)(46 55)(47 52)(48 49)(57 80)(58 77)(59 74)(60 79)(61 76)(62 73)(63 78)(64 75)
G:=sub<Sym(80)| (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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,45,21,41)(18,44,22,48)(19,43,23,47)(20,42,24,46)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,66,12,41,75)(2,67,13,42,76)(3,68,14,43,77)(4,69,15,44,78)(5,70,16,45,79)(6,71,9,46,80)(7,72,10,47,73)(8,65,11,48,74)(17,29,60,50,40)(18,30,61,51,33)(19,31,62,52,34)(20,32,63,53,35)(21,25,64,54,36)(22,26,57,55,37)(23,27,58,56,38)(24,28,59,49,39), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,37)(10,34)(11,39)(12,36)(13,33)(14,38)(15,35)(16,40)(17,70)(18,67)(19,72)(20,69)(21,66)(22,71)(23,68)(24,65)(41,54)(42,51)(43,56)(44,53)(45,50)(46,55)(47,52)(48,49)(57,80)(58,77)(59,74)(60,79)(61,76)(62,73)(63,78)(64,75)>;
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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,45,21,41)(18,44,22,48)(19,43,23,47)(20,42,24,46)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,66,12,41,75)(2,67,13,42,76)(3,68,14,43,77)(4,69,15,44,78)(5,70,16,45,79)(6,71,9,46,80)(7,72,10,47,73)(8,65,11,48,74)(17,29,60,50,40)(18,30,61,51,33)(19,31,62,52,34)(20,32,63,53,35)(21,25,64,54,36)(22,26,57,55,37)(23,27,58,56,38)(24,28,59,49,39), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,37)(10,34)(11,39)(12,36)(13,33)(14,38)(15,35)(16,40)(17,70)(18,67)(19,72)(20,69)(21,66)(22,71)(23,68)(24,65)(41,54)(42,51)(43,56)(44,53)(45,50)(46,55)(47,52)(48,49)(57,80)(58,77)(59,74)(60,79)(61,76)(62,73)(63,78)(64,75) );
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),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,60,5,64),(2,59,6,63),(3,58,7,62),(4,57,8,61),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(17,45,21,41),(18,44,22,48),(19,43,23,47),(20,42,24,46),(25,75,29,79),(26,74,30,78),(27,73,31,77),(28,80,32,76),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)], [(1,66,12,41,75),(2,67,13,42,76),(3,68,14,43,77),(4,69,15,44,78),(5,70,16,45,79),(6,71,9,46,80),(7,72,10,47,73),(8,65,11,48,74),(17,29,60,50,40),(18,30,61,51,33),(19,31,62,52,34),(20,32,63,53,35),(21,25,64,54,36),(22,26,57,55,37),(23,27,58,56,38),(24,28,59,49,39)], [(1,25),(2,30),(3,27),(4,32),(5,29),(6,26),(7,31),(8,28),(9,37),(10,34),(11,39),(12,36),(13,33),(14,38),(15,35),(16,40),(17,70),(18,67),(19,72),(20,69),(21,66),(22,71),(23,68),(24,65),(41,54),(42,51),(43,56),(44,53),(45,50),(46,55),(47,52),(48,49),(57,80),(58,77),(59,74),(60,79),(61,76),(62,73),(63,78),(64,75)]])
Q16⋊D5 is a maximal subgroup of
D20.30D4 Q16⋊D10 D8⋊11D10 D5×C8.C22 D40⋊C22 C40.C23 D20.44D4 C24.2D10 D30.3D4 D20.13D6 C60.C23 C60.39C23 D20.17D6 Q16⋊D15 CSU2(𝔽3)⋊D5 D10.2S4
Q16⋊D5 is a maximal quotient of
C5⋊Q16⋊5C4 Q8⋊Dic10 Q8⋊C4⋊D5 C40⋊8C4.C2 Dic10.11D4 Q8.2Dic10 (Q8×D5)⋊C4 Q8⋊(C4×D5) D10.11SD16 Q8⋊2D20 Q8.D20 C5⋊(C8⋊D4) D10⋊1C8.C2 C5⋊2C8.D4 Q8⋊D5⋊6C4 Dic5⋊SD16 C40⋊4Q8 Dic10.2Q8 C40⋊20(C2×C4) D10.13D8 C8⋊3D20 C2.D8⋊7D5 C40⋊21(C2×C4) D20⋊2Q8 Dic5⋊3Q16 Q16⋊Dic5 (C2×Q16)⋊D5 D10⋊5Q16 D20.17D4 C40.36D4 C40.37D4 C24.2D10 D30.3D4 D20.13D6 C60.C23 C60.39C23 D20.17D6 Q16⋊D15
Matrix representation of Q16⋊D5 ►in GL4(𝔽41) generated by
| 0 | 0 | 39 | 17 |
| 0 | 0 | 24 | 2 |
| 1 | 12 | 39 | 17 |
| 29 | 40 | 24 | 2 |
| 17 | 11 | 2 | 0 |
| 30 | 22 | 0 | 2 |
| 18 | 11 | 24 | 30 |
| 30 | 23 | 11 | 19 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 |
| 30 | 22 | 0 | 2 |
| 17 | 11 | 2 | 0 |
| 0 | 40 | 30 | 24 |
| 40 | 0 | 19 | 11 |
G:=sub<GL(4,GF(41))| [0,0,1,29,0,0,12,40,39,24,39,24,17,2,17,2],[17,30,18,30,11,22,11,23,2,0,24,11,0,2,30,19],[0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,34],[30,17,0,40,22,11,40,0,0,2,30,19,2,0,24,11] >;
Q16⋊D5 in GAP, Magma, Sage, TeX
Q_{16}\rtimes D_5 % in TeX
G:=Group("Q16:D5"); // GroupNames label
G:=SmallGroup(160,139);
// by ID
G=gap.SmallGroup(160,139);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,116,86,297,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^5=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,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations
Export