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G = D5×D16order 320 = 26·5

Direct product of D5 and D16

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

Aliases: D5×D16, D804C2, D81D10, C164D10, C802C22, D10.24D8, D405C22, Dic5.7D8, C40.13C23, C52(C2×D16), (D5×D8)⋊3C2, C4.1(D4×D5), (D5×C16)⋊1C2, (C5×D16)⋊2C2, C5⋊D161C2, C2.16(D5×D8), C20.7(C2×D4), (C4×D5).57D4, C10.32(C2×D8), C52C8.23D4, (C5×D8)⋊5C22, C52C165C22, C8.19(C22×D5), (C8×D5).38C22, SmallGroup(320,537)

Series: Derived Chief Lower central Upper central

C1C40 — D5×D16
C1C5C10C20C40C8×D5D5×D8 — D5×D16
C5C10C20C40 — D5×D16
C1C2C4C8D16

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 [×6], C4, C4, C22 [×9], C5, C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C16, C16, C2×C8, D8 [×2], D8 [×4], C2×D4 [×2], Dic5, C20, D10, D10 [×6], C2×C10 [×2], C2×C16, D16, D16 [×3], C2×D8 [×2], C52C8, C40, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×2], C2×D16, C52C16, C80, C8×D5, D40 [×2], D4⋊D5 [×2], C5×D8 [×2], D4×D5 [×2], D5×C16, D80, C5⋊D16 [×2], C5×D16, D5×D8 [×2], D5×D16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], D16 [×2], 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 47 61 69 22)(2 48 62 70 23)(3 33 63 71 24)(4 34 64 72 25)(5 35 49 73 26)(6 36 50 74 27)(7 37 51 75 28)(8 38 52 76 29)(9 39 53 77 30)(10 40 54 78 31)(11 41 55 79 32)(12 42 56 80 17)(13 43 57 65 18)(14 44 58 66 19)(15 45 59 67 20)(16 46 60 68 21)
(1 30)(2 31)(3 32)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(33 79)(34 80)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 24)(21 23)(28 32)(29 31)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 48)(49 57)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(65 73)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B8A8B8C8D10A10B10C10D10E10F16A16B16C16D16E16F16G16H20A20B40A40B40C40D80A···80H
order1222222244558888101010101010161616161616161620204040404080···80
size11558840402102222101022161616162222101010104444444···4

44 irreducible representations

dim11111122222222444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8D8D10D10D16D4×D5D5×D8D5×D16
kernelD5×D16D5×C16D80C5⋊D16C5×D16D5×D8C52C8C4×D5D16Dic5D10C16D8D5C4C2C1
# reps11121211222248248

Matrix representation of D5×D16 in GL4(𝔽241) generated by

1000
0100
002401
0050190
,
240000
024000
002400
00501
,
5817300
9212900
002400
000240
,
05700
148000
0010
0001
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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