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## G = D5×D16order 320 = 26·5

### Direct product of D5 and D16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D5×D16
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×D8 — D5×D16
 Lower central C5 — C10 — C20 — C40 — D5×D16
 Upper central C1 — C2 — C4 — C8 — D16

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, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C2×D16, C52C16, 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

Smallest permutation representation of 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

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