metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D16⋊2D5, D8⋊2D10, C16⋊2D10, C80⋊4C22, D10.14D8, C40.14C23, Dic5.16D8, D40.1C22, Dic20⋊4C22, (D5×D8)⋊4C2, C4.2(D4×D5), (C5×D16)⋊4C2, C5⋊D16⋊2C2, D8.D5⋊1C2, C16⋊D5⋊3C2, (C4×D5).7D4, C80⋊C2⋊3C2, C2.17(D5×D8), C20.8(C2×D4), C5⋊2C8.2D4, D8⋊3D5⋊3C2, C5⋊2(C16⋊C22), C10.33(C2×D8), (C5×D8)⋊6C22, C5⋊2C16⋊1C22, (C8×D5).3C22, C8.20(C22×D5), SmallGroup(320,538)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D16⋊D5
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, bd=db, dcd=c-1 >
Subgroups: 534 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, M5(2), D16, D16, SD32, C2×D8, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C16⋊C22, C5⋊2C16, C80, C8×D5, D40, Dic20, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, C80⋊C2, C16⋊D5, C5⋊D16, D8.D5, C5×D16, D5×D8, D8⋊3D5, D16⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, C16⋊C22, D4×D5, D5×D8, D16⋊D5
►On 80 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)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(30 32)(33 35)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(49 57)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
(1 23 34 65 53)(2 24 35 66 54)(3 25 36 67 55)(4 26 37 68 56)(5 27 38 69 57)(6 28 39 70 58)(7 29 40 71 59)(8 30 41 72 60)(9 31 42 73 61)(10 32 43 74 62)(11 17 44 75 63)(12 18 45 76 64)(13 19 46 77 49)(14 20 47 78 50)(15 21 48 79 51)(16 22 33 80 52)
(1 53)(2 62)(3 55)(4 64)(5 57)(6 50)(7 59)(8 52)(9 61)(10 54)(11 63)(12 56)(13 49)(14 58)(15 51)(16 60)(17 75)(18 68)(19 77)(20 70)(21 79)(22 72)(23 65)(24 74)(25 67)(26 76)(27 69)(28 78)(29 71)(30 80)(31 73)(32 66)(33 41)(35 43)(37 45)(39 47)
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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (1,23,34,65,53)(2,24,35,66,54)(3,25,36,67,55)(4,26,37,68,56)(5,27,38,69,57)(6,28,39,70,58)(7,29,40,71,59)(8,30,41,72,60)(9,31,42,73,61)(10,32,43,74,62)(11,17,44,75,63)(12,18,45,76,64)(13,19,46,77,49)(14,20,47,78,50)(15,21,48,79,51)(16,22,33,80,52), (1,53)(2,62)(3,55)(4,64)(5,57)(6,50)(7,59)(8,52)(9,61)(10,54)(11,63)(12,56)(13,49)(14,58)(15,51)(16,60)(17,75)(18,68)(19,77)(20,70)(21,79)(22,72)(23,65)(24,74)(25,67)(26,76)(27,69)(28,78)(29,71)(30,80)(31,73)(32,66)(33,41)(35,43)(37,45)(39,47)>;
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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (1,23,34,65,53)(2,24,35,66,54)(3,25,36,67,55)(4,26,37,68,56)(5,27,38,69,57)(6,28,39,70,58)(7,29,40,71,59)(8,30,41,72,60)(9,31,42,73,61)(10,32,43,74,62)(11,17,44,75,63)(12,18,45,76,64)(13,19,46,77,49)(14,20,47,78,50)(15,21,48,79,51)(16,22,33,80,52), (1,53)(2,62)(3,55)(4,64)(5,57)(6,50)(7,59)(8,52)(9,61)(10,54)(11,63)(12,56)(13,49)(14,58)(15,51)(16,60)(17,75)(18,68)(19,77)(20,70)(21,79)(22,72)(23,65)(24,74)(25,67)(26,76)(27,69)(28,78)(29,71)(30,80)(31,73)(32,66)(33,41)(35,43)(37,45)(39,47) );
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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(30,32),(33,35),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(49,57),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)], [(1,23,34,65,53),(2,24,35,66,54),(3,25,36,67,55),(4,26,37,68,56),(5,27,38,69,57),(6,28,39,70,58),(7,29,40,71,59),(8,30,41,72,60),(9,31,42,73,61),(10,32,43,74,62),(11,17,44,75,63),(12,18,45,76,64),(13,19,46,77,49),(14,20,47,78,50),(15,21,48,79,51),(16,22,33,80,52)], [(1,53),(2,62),(3,55),(4,64),(5,57),(6,50),(7,59),(8,52),(9,61),(10,54),(11,63),(12,56),(13,49),(14,58),(15,51),(16,60),(17,75),(18,68),(19,77),(20,70),(21,79),(22,72),(23,65),(24,74),(25,67),(26,76),(27,69),(28,78),(29,71),(30,80),(31,73),(32,66),(33,41),(35,43),(37,45),(39,47)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 10E | 10F | 16A | 16B | 16C | 16D | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 8 | 10 | 40 | 2 | 10 | 40 | 2 | 2 | 2 | 2 | 20 | 2 | 2 | 16 | 16 | 16 | 16 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | C16⋊C22 | D4×D5 | D5×D8 | D16⋊D5 |
| kernel | D16⋊D5 | C80⋊C2 | C16⋊D5 | C5⋊D16 | D8.D5 | C5×D16 | D5×D8 | D8⋊3D5 | C5⋊2C8 | C4×D5 | D16 | Dic5 | D10 | C16 | D8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 |
Matrix representation of D16⋊D5 ►in GL4(𝔽241) generated by
| 109 | 32 | 11 | 158 |
| 209 | 132 | 83 | 230 |
| 115 | 162 | 120 | 190 |
| 79 | 126 | 51 | 121 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 240 | 0 |
| 0 | 1 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 240 | 189 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 189 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [109,209,115,79,32,132,162,126,11,83,120,51,158,230,190,121],[1,0,1,0,0,1,0,1,0,0,240,0,0,0,0,240],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
D16⋊D5 in GAP, Magma, Sage, TeX
D_{16}\rtimes D_5 % in TeX
G:=Group("D16:D5"); // GroupNames label
G:=SmallGroup(320,538);
// by ID
G=gap.SmallGroup(320,538);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,135,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations