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G = D5xQ16  order 160 = 25·5

Direct product of D5 and Q16

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

Aliases: D5xQ16, C8.9D10, Q8.3D10, Dic20:5C2, D10.25D4, C20.8C23, C40.7C22, Dic5.9D4, Dic10.4C22, C5:2(C2xQ16), (C5xQ16):2C2, C5:Q16:3C2, (C8xD5).1C2, C2.22(D4xD5), (Q8xD5).1C2, C10.34(C2xD4), C4.8(C22xD5), C5:2C8.7C22, (C5xQ8).3C22, (C4xD5).19C22, SmallGroup(160,138)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D5xQ16
Chief series
C1 — C5 — C10 — C20 — C4xD5 — Q8xD5 — D5xQ16
Lower central
C5 — C10 — C20 — D5xQ16
Upper central
C1 — C2 — C4 — Q16

Generators and relations for D5xQ16
 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, C2xC4, Q8, Q8, D5, C10, C2xC8, Q16, Q16, C2xQ8, Dic5, Dic5, C20, C20, D10, C2xQ16, C5:2C8, C40, Dic10, Dic10, C4xD5, C4xD5, C5xQ8, C8xD5, Dic20, C5:Q16, C5xQ16, Q8xD5, D5xQ16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2xD4, D10, C2xQ16, C22xD5, D4xD5, D5xQ16

Character table of D5xQ16

 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
ρ192-2-2200000022√2-√2√2-√2-2-2000000√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ202-22-200000022√2-√2-√2√2-2-2000000√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ212-22-200000022-√2√2√2-√2-2-2000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ222-2-2200000022-√2√2-√2√2-2-2000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ234400-400000-1+√5-1-√50000-1+√5-1-√51+√51-√500000000    orthogonal lifted from D4xD5
ρ244400-400000-1-√5-1+√50000-1-√5-1+√51-√51+√500000000    orthogonal lifted from D4xD5
ρ254-400000000-1+√5-1-√5-2√22√2001-√51+√5000000-ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5-ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52    symplectic faithful, Schur index 2
ρ264-400000000-1+√5-1-√52√2-2√2001-√51+√5000000-ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5-ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52    symplectic faithful, Schur index 2
ρ274-400000000-1-√5-1+√5-2√22√2001+√51-√5000000ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52-ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5-ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5    symplectic faithful, Schur index 2
ρ284-400000000-1-√5-1+√52√2-2√2001+√51-√5000000-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52-ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5-ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5    symplectic faithful, Schur index 2

Smallest permutation representation of D5xQ16
►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)]])
 

D5xQ16 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
D5xQ16 is a maximal quotient of
 Dic5:4Q16  Dic5.3Q16  Dic5:Q16  Dic5.9Q16  D10:4Q16  D10.7Q16  D10:Q16  Dic5:5Q16  C40:2Q8  Dic10:2Q8  D10.8Q16  D10:2Q16  C40.26D4  Dic5:3Q16  D10:5Q16  D10:3Q16  Dic10.D6  D15:Q16

Matrix representation of D5xQ16 ►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] >;
 

D5xQ16 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 D5xQ16 in TeX

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