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G = D5×Q16order 160 = 25·5

Direct product of D5 and Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×Q16, C8.9D10, Q8.3D10, Dic205C2, D10.25D4, C20.8C23, C40.7C22, Dic5.9D4, Dic10.4C22, C52(C2×Q16), (C5×Q16)⋊2C2, C5⋊Q163C2, (C8×D5).1C2, C2.22(D4×D5), (Q8×D5).1C2, C10.34(C2×D4), C4.8(C22×D5), C52C8.7C22, (C5×Q8).3C22, (C4×D5).19C22, SmallGroup(160,138)

Series: Derived Chief Lower central Upper central

C1C20 — D5×Q16
C1C5C10C20C4×D5Q8×D5 — D5×Q16
C5C10C20 — D5×Q16
C1C2C4Q16

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 12A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A10B20A20B20C20D20E20F40A40B40C40D
 size 115524410202022221010224488884444
ρ11111111111111111111111111111    trivial
ρ2111111-111-111-1-1-1-11111-111-1-1-1-1-1    linear of order 2
ρ311111-111-1111-1-1-1-111111-1-11-1-1-1-1    linear of order 2
ρ411111-1-11-1-11111111111-1-1-1-11111    linear of order 2
ρ511-1-11-11-11-111-1-11111111-1-11-1-1-1-1    linear of order 2
ρ611-1-11-1-1-1111111-1-11111-1-1-1-11111    linear of order 2
ρ711-1-1111-1-1-11111-1-1111111111111    linear of order 2
ρ811-1-111-1-1-1111-1-1111111-111-1-1-1-1-1    linear of order 2
ρ92222-200-20022000022-2-200000000    orthogonal lifted from D4
ρ1022-2-2-20020022000022-2-200000000    orthogonal lifted from D4
ρ1122002-2-2000-1+5/2-1-5/22200-1+5/2-1-5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ122200222000-1+5/2-1-5/22200-1+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
ρ1322002-22000-1+5/2-1-5/2-2-200-1+5/2-1-5/2-1-5/2-1+5/2-1-5/21+5/21-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ14220022-2000-1+5/2-1-5/2-2-200-1+5/2-1-5/2-1-5/2-1+5/21+5/2-1-5/2-1+5/21-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1522002-22000-1-5/2-1+5/2-2-200-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21-5/21+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ162200222000-1-5/2-1+5/22200-1-5/2-1+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
ρ1722002-2-2000-1-5/2-1+5/22200-1-5/2-1+5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ18220022-2000-1-5/2-1+5/2-2-200-1-5/2-1+5/2-1+5/2-1-5/21-5/2-1+5/2-1-5/21+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ192-2-22000000222-22-2-2-20000002-2-22    symplectic lifted from Q16, Schur index 2
ρ202-22-2000000222-2-22-2-20000002-2-22    symplectic lifted from Q16, Schur index 2
ρ212-22-200000022-222-2-2-2000000-222-2    symplectic lifted from Q16, Schur index 2
ρ222-2-2200000022-22-22-2-2000000-222-2    symplectic lifted from Q16, Schur index 2
ρ234400-400000-1+5-1-50000-1+5-1-51+51-500000000    orthogonal lifted from D4×D5
ρ244400-400000-1-5-1+50000-1-5-1+51-51+500000000    orthogonal lifted from D4×D5
ρ254-400000000-1+5-1-5-2222001-51+500000087ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ264-400000000-1+5-1-522-22001-51+500000083ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ274-400000000-1-5-1+5-2222001+51-5000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ5    symplectic faithful, Schur index 2
ρ284-400000000-1-5-1+522-22001+51-500000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5    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

34100
40000
0010
0001
,
13400
04000
00400
00040
,
1000
0100
0030
00014
,
1000
0100
0001
00400
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

Export

Character table of D5×Q16 in TeX

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