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## G = D16⋊D5order 320 = 26·5

### 2nd semidirect product of D16 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

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

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 [×4], C4, C4 [×2], C22 [×6], C5, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], C16, C16, C2×C8, D8 [×2], D8 [×2], SD16, Q16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], M5(2), D16, D16, SD32 [×2], C2×D8, C4○D8, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C16⋊C22, C52C16, C80, C8×D5, D40, Dic20, D4⋊D5, D4.D5, C5×D8 [×2], D4×D5, D42D5, C80⋊C2, C16⋊D5, C5⋊D16, D8.D5, C5×D16, D5×D8, D83D5, D16⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C22×D5, C16⋊C22, D4×D5, D5×D8, D16⋊D5

Smallest permutation representation of D16⋊D5
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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 21)(18 20)(22 32)(23 31)(24 30)(25 29)(26 28)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(49 59)(50 58)(51 57)(52 56)(53 55)(60 64)(61 63)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(78 80)
(1 71 40 27 54)(2 72 41 28 55)(3 73 42 29 56)(4 74 43 30 57)(5 75 44 31 58)(6 76 45 32 59)(7 77 46 17 60)(8 78 47 18 61)(9 79 48 19 62)(10 80 33 20 63)(11 65 34 21 64)(12 66 35 22 49)(13 67 36 23 50)(14 68 37 24 51)(15 69 38 25 52)(16 70 39 26 53)
(1 54)(2 63)(3 56)(4 49)(5 58)(6 51)(7 60)(8 53)(9 62)(10 55)(11 64)(12 57)(13 50)(14 59)(15 52)(16 61)(17 77)(18 70)(19 79)(20 72)(21 65)(22 74)(23 67)(24 76)(25 69)(26 78)(27 71)(28 80)(29 73)(30 66)(31 75)(32 68)(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)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(78,80), (1,71,40,27,54)(2,72,41,28,55)(3,73,42,29,56)(4,74,43,30,57)(5,75,44,31,58)(6,76,45,32,59)(7,77,46,17,60)(8,78,47,18,61)(9,79,48,19,62)(10,80,33,20,63)(11,65,34,21,64)(12,66,35,22,49)(13,67,36,23,50)(14,68,37,24,51)(15,69,38,25,52)(16,70,39,26,53), (1,54)(2,63)(3,56)(4,49)(5,58)(6,51)(7,60)(8,53)(9,62)(10,55)(11,64)(12,57)(13,50)(14,59)(15,52)(16,61)(17,77)(18,70)(19,79)(20,72)(21,65)(22,74)(23,67)(24,76)(25,69)(26,78)(27,71)(28,80)(29,73)(30,66)(31,75)(32,68)(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)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(78,80), (1,71,40,27,54)(2,72,41,28,55)(3,73,42,29,56)(4,74,43,30,57)(5,75,44,31,58)(6,76,45,32,59)(7,77,46,17,60)(8,78,47,18,61)(9,79,48,19,62)(10,80,33,20,63)(11,65,34,21,64)(12,66,35,22,49)(13,67,36,23,50)(14,68,37,24,51)(15,69,38,25,52)(16,70,39,26,53), (1,54)(2,63)(3,56)(4,49)(5,58)(6,51)(7,60)(8,53)(9,62)(10,55)(11,64)(12,57)(13,50)(14,59)(15,52)(16,61)(17,77)(18,70)(19,79)(20,72)(21,65)(22,74)(23,67)(24,76)(25,69)(26,78)(27,71)(28,80)(29,73)(30,66)(31,75)(32,68)(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),(17,21),(18,20),(22,32),(23,31),(24,30),(25,29),(26,28),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(49,59),(50,58),(51,57),(52,56),(53,55),(60,64),(61,63),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(78,80)], [(1,71,40,27,54),(2,72,41,28,55),(3,73,42,29,56),(4,74,43,30,57),(5,75,44,31,58),(6,76,45,32,59),(7,77,46,17,60),(8,78,47,18,61),(9,79,48,19,62),(10,80,33,20,63),(11,65,34,21,64),(12,66,35,22,49),(13,67,36,23,50),(14,68,37,24,51),(15,69,38,25,52),(16,70,39,26,53)], [(1,54),(2,63),(3,56),(4,49),(5,58),(6,51),(7,60),(8,53),(9,62),(10,55),(11,64),(12,57),(13,50),(14,59),(15,52),(16,61),(17,77),(18,70),(19,79),(20,72),(21,65),(22,74),(23,67),(24,76),(25,69),(26,78),(27,71),(28,80),(29,73),(30,66),(31,75),(32,68),(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

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