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 [×6], C4, C4, C22 [×9], C5, C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C16, C16, C2×C8, D8 [×2], D8 [×4], C2×D4 [×2], Dic5, C20, D10, D10 [×6], C2×C10 [×2], C2×C16, D16, D16 [×3], C2×D8 [×2], C5⋊2C8, C40, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×2], C2×D16, C5⋊2C16, C80, C8×D5, D40 [×2], D4⋊D5 [×2], C5×D8 [×2], D4×D5 [×2], D5×C16, D80, C5⋊D16 [×2], C5×D16, D5×D8 [×2], D5×D16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], D16 [×2], C2×D8, C22×D5, C2×D16, D4×D5, D5×D8, D5×D16
►On 80 points
(1 47 61 69 22)(2 48 62 70 23)(3 33 63 71 24)(4 34 64 72 25)(5 35 49 73 26)(6 36 50 74 27)(7 37 51 75 28)(8 38 52 76 29)(9 39 53 77 30)(10 40 54 78 31)(11 41 55 79 32)(12 42 56 80 17)(13 43 57 65 18)(14 44 58 66 19)(15 45 59 67 20)(16 46 60 68 21)
(1 30)(2 31)(3 32)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(33 79)(34 80)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 24)(21 23)(28 32)(29 31)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 48)(49 57)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(65 73)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)
G:=sub<Sym(80)| (1,47,61,69,22)(2,48,62,70,23)(3,33,63,71,24)(4,34,64,72,25)(5,35,49,73,26)(6,36,50,74,27)(7,37,51,75,28)(8,38,52,76,29)(9,39,53,77,30)(10,40,54,78,31)(11,41,55,79,32)(12,42,56,80,17)(13,43,57,65,18)(14,44,58,66,19)(15,45,59,67,20)(16,46,60,68,21), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(33,79)(34,80)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,24)(21,23)(28,32)(29,31)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,48)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(65,73)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)>;
G:=Group( (1,47,61,69,22)(2,48,62,70,23)(3,33,63,71,24)(4,34,64,72,25)(5,35,49,73,26)(6,36,50,74,27)(7,37,51,75,28)(8,38,52,76,29)(9,39,53,77,30)(10,40,54,78,31)(11,41,55,79,32)(12,42,56,80,17)(13,43,57,65,18)(14,44,58,66,19)(15,45,59,67,20)(16,46,60,68,21), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(33,79)(34,80)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,24)(21,23)(28,32)(29,31)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,48)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(65,73)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78) );
G=PermutationGroup([(1,47,61,69,22),(2,48,62,70,23),(3,33,63,71,24),(4,34,64,72,25),(5,35,49,73,26),(6,36,50,74,27),(7,37,51,75,28),(8,38,52,76,29),(9,39,53,77,30),(10,40,54,78,31),(11,41,55,79,32),(12,42,56,80,17),(13,43,57,65,18),(14,44,58,66,19),(15,45,59,67,20),(16,46,60,68,21)], [(1,30),(2,31),(3,32),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(33,79),(34,80),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,27),(18,26),(19,25),(20,24),(21,23),(28,32),(29,31),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(46,48),(49,57),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62),(65,73),(66,72),(67,71),(68,70),(74,80),(75,79),(76,78)])
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