direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D16, D80⋊4C2, D8⋊1D10, C16⋊4D10, C80⋊2C22, D10.24D8, D40⋊5C22, Dic5.7D8, C40.13C23, C5⋊2(C2×D16), (D5×D8)⋊3C2, C4.1(D4×D5), (D5×C16)⋊1C2, (C5×D16)⋊2C2, C5⋊D16⋊1C2, C2.16(D5×D8), C20.7(C2×D4), (C4×D5).57D4, C10.32(C2×D8), C5⋊2C8.23D4, (C5×D8)⋊5C22, C5⋊2C16⋊5C22, C8.19(C22×D5), (C8×D5).38C22, SmallGroup(320,537)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D16
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 678 in 98 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, C23, D5, D5, C10, C10, C16, C16, C2×C8, D8, D8, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C16, D16, D16, C2×D8, C5⋊2C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C2×D16, C5⋊2C16, C80, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, D5×C16, D80, C5⋊D16, C5×D16, D5×D8, D5×D16
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, D16, C2×D8, C22×D5, C2×D16, D4×D5, D5×D8, D5×D16
►On 80 points
Generators in S80
(1 25 34 55 71)(2 26 35 56 72)(3 27 36 57 73)(4 28 37 58 74)(5 29 38 59 75)(6 30 39 60 76)(7 31 40 61 77)(8 32 41 62 78)(9 17 42 63 79)(10 18 43 64 80)(11 19 44 49 65)(12 20 45 50 66)(13 21 46 51 67)(14 22 47 52 68)(15 23 48 53 69)(16 24 33 54 70)
(1 79)(2 80)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(33 35)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(62 64)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(78 80)
G:=sub<Sym(80)| (1,25,34,55,71)(2,26,35,56,72)(3,27,36,57,73)(4,28,37,58,74)(5,29,38,59,75)(6,30,39,60,76)(7,31,40,61,77)(8,32,41,62,78)(9,17,42,63,79)(10,18,43,64,80)(11,19,44,49,65)(12,20,45,50,66)(13,21,46,51,67)(14,22,47,52,68)(15,23,48,53,69)(16,24,33,54,70), (1,79)(2,80)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(62,64)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(78,80)>;
G:=Group( (1,25,34,55,71)(2,26,35,56,72)(3,27,36,57,73)(4,28,37,58,74)(5,29,38,59,75)(6,30,39,60,76)(7,31,40,61,77)(8,32,41,62,78)(9,17,42,63,79)(10,18,43,64,80)(11,19,44,49,65)(12,20,45,50,66)(13,21,46,51,67)(14,22,47,52,68)(15,23,48,53,69)(16,24,33,54,70), (1,79)(2,80)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(62,64)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(78,80) );
G=PermutationGroup([[(1,25,34,55,71),(2,26,35,56,72),(3,27,36,57,73),(4,28,37,58,74),(5,29,38,59,75),(6,30,39,60,76),(7,31,40,61,77),(8,32,41,62,78),(9,17,42,63,79),(10,18,43,64,80),(11,19,44,49,65),(12,20,45,50,66),(13,21,46,51,67),(14,22,47,52,68),(15,23,48,53,69),(16,24,33,54,70)], [(1,79),(2,80),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(33,35),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(62,64),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(78,80)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 5 | 5 | 8 | 8 | 40 | 40 | 2 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D16 | D4×D5 | D5×D8 | D5×D16 |
| kernel | D5×D16 | D5×C16 | D80 | C5⋊D16 | C5×D16 | D5×D8 | C5⋊2C8 | C4×D5 | D16 | Dic5 | D10 | C16 | D8 | D5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of D5×D16 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 50 | 190 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 50 | 1 |
| 58 | 173 | 0 | 0 |
| 92 | 129 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 57 | 0 | 0 |
| 148 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[240,0,0,0,0,240,0,0,0,0,240,50,0,0,0,1],[58,92,0,0,173,129,0,0,0,0,240,0,0,0,0,240],[0,148,0,0,57,0,0,0,0,0,1,0,0,0,0,1] >;
D5×D16 in GAP, Magma, Sage, TeX
D_5\times D_{16} % in TeX
G:=Group("D5xD16"); // GroupNames label
G:=SmallGroup(320,537);
// by ID
G=gap.SmallGroup(320,537);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,135,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations