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## G = D5×Q16order 160 = 25·5

### Direct product of D5 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5×Q16
 Chief series C1 — C5 — C10 — C20 — C4×D5 — Q8×D5 — D5×Q16
 Lower central C5 — C10 — C20 — D5×Q16
 Upper central C1 — C2 — C4 — Q16

Generators and relations for D5×Q16
G = < a,b,c,d | a5=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 184 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, Q8, Q8, D5, C10, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×Q16, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C8×D5, Dic20, C5⋊Q16, C5×Q16, Q8×D5, D5×Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C2×Q16, C22×D5, D4×D5, D5×Q16

Character table of D5×Q16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A 10B 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D size 1 1 5 5 2 4 4 10 20 20 2 2 2 2 10 10 2 2 4 4 8 8 8 8 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 -2 0 0 -2 0 0 2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 -2 0 0 2 0 0 2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 0 0 2 -2 -2 0 0 0 -1+√5/2 -1-√5/2 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ12 2 2 0 0 2 2 2 0 0 0 -1+√5/2 -1-√5/2 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ13 2 2 0 0 2 -2 2 0 0 0 -1+√5/2 -1-√5/2 -2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ14 2 2 0 0 2 2 -2 0 0 0 -1+√5/2 -1-√5/2 -2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ15 2 2 0 0 2 -2 2 0 0 0 -1-√5/2 -1+√5/2 -2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ16 2 2 0 0 2 2 2 0 0 0 -1-√5/2 -1+√5/2 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ17 2 2 0 0 2 -2 -2 0 0 0 -1-√5/2 -1+√5/2 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ18 2 2 0 0 2 2 -2 0 0 0 -1-√5/2 -1+√5/2 -2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ19 2 -2 -2 2 0 0 0 0 0 0 2 2 √2 -√2 √2 -√2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ20 2 -2 2 -2 0 0 0 0 0 0 2 2 √2 -√2 -√2 √2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ21 2 -2 2 -2 0 0 0 0 0 0 2 2 -√2 √2 √2 -√2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ22 2 -2 -2 2 0 0 0 0 0 0 2 2 -√2 √2 -√2 √2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ23 4 4 0 0 -4 0 0 0 0 0 -1+√5 -1-√5 0 0 0 0 -1+√5 -1-√5 1+√5 1-√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ24 4 4 0 0 -4 0 0 0 0 0 -1-√5 -1+√5 0 0 0 0 -1-√5 -1+√5 1-√5 1+√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ25 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 -2√2 2√2 0 0 1-√5 1+√5 0 0 0 0 0 0 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 symplectic faithful, Schur index 2 ρ26 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 2√2 -2√2 0 0 1-√5 1+√5 0 0 0 0 0 0 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 -2√2 2√2 0 0 1+√5 1-√5 0 0 0 0 0 0 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 symplectic faithful, Schur index 2 ρ28 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 2√2 -2√2 0 0 1+√5 1-√5 0 0 0 0 0 0 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 symplectic faithful, Schur index 2

Smallest permutation representation of D5×Q16
On 80 points
Generators in S80
(1 66 62 39 15)(2 67 63 40 16)(3 68 64 33 9)(4 69 57 34 10)(5 70 58 35 11)(6 71 59 36 12)(7 72 60 37 13)(8 65 61 38 14)(17 29 75 50 45)(18 30 76 51 46)(19 31 77 52 47)(20 32 78 53 48)(21 25 79 54 41)(22 26 80 55 42)(23 27 73 56 43)(24 28 74 49 44)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 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)
(1 75 5 79)(2 74 6 78)(3 73 7 77)(4 80 8 76)(9 27 13 31)(10 26 14 30)(11 25 15 29)(12 32 16 28)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)

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

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

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

D5×Q16 is a maximal subgroup of
D5.Q32  Dic20.C4  SD32⋊D5  Q32⋊D5  Dic20⋊C4  D20.30D4  D20.47D4  D20.44D4  Dic10.D6  D15⋊Q16
D5×Q16 is a maximal quotient of
Dic54Q16  Dic5.3Q16  Dic5⋊Q16  Dic5.9Q16  D104Q16  D10.7Q16  D10⋊Q16  Dic55Q16  C402Q8  Dic102Q8  D10.8Q16  D102Q16  C40.26D4  Dic53Q16  D105Q16  D103Q16  Dic10.D6  D15⋊Q16

Matrix representation of D5×Q16 in GL4(𝔽41) generated by

 34 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 1 34 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 14
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 40 0
G:=sub<GL(4,GF(41))| [34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,14],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0] >;

D5×Q16 in GAP, Magma, Sage, TeX

D_5\times Q_{16}
% in TeX

G:=Group("D5xQ16");
// GroupNames label

G:=SmallGroup(160,138);
// by ID

G=gap.SmallGroup(160,138);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,116,86,297,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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