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G = D5xD16order 320 = 26·5

Direct product of D5 and D16

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

Aliases: D5xD16, D80:4C2, D8:1D10, C16:4D10, C80:2C22, D10.24D8, D40:5C22, Dic5.7D8, C40.13C23, C5:2(C2xD16), (D5xD8):3C2, C4.1(D4xD5), (D5xC16):1C2, (C5xD16):2C2, C5:D16:1C2, C2.16(D5xD8), C20.7(C2xD4), (C4xD5).57D4, C10.32(C2xD8), C5:2C8.23D4, (C5xD8):5C22, C5:2C16:5C22, C8.19(C22xD5), (C8xD5).38C22, SmallGroup(320,537)

Series: Derived Chief Lower central Upper central

C1C40 — D5xD16
C1C5C10C20C40C8xD5D5xD8 — D5xD16
C5C10C20C40 — D5xD16
C1C2C4C8D16

Generators and relations for D5xD16
 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, C4, C4, C22, C5, C8, C8, C2xC4, D4, C23, D5, D5, C10, C10, C16, C16, C2xC8, D8, D8, C2xD4, Dic5, C20, D10, D10, C2xC10, C2xC16, D16, D16, C2xD8, C5:2C8, C40, C4xD5, D20, C5:D4, C5xD4, C22xD5, C2xD16, C5:2C16, C80, C8xD5, D40, D4:D5, C5xD8, D4xD5, D5xC16, D80, C5:D16, C5xD16, D5xD8, D5xD16
Quotients: C1, C2, C22, D4, C23, D5, D8, C2xD4, D10, D16, C2xD8, C22xD5, C2xD16, D4xD5, D5xD8, D5xD16

Smallest permutation representation of D5xD16
On 80 points
Generators in S80
(1 25 34 55 71)(2 26 35 56 72)(3 27 36 57 73)(4 28 37 58 74)(5 29 38 59 75)(6 30 39 60 76)(7 31 40 61 77)(8 32 41 62 78)(9 17 42 63 79)(10 18 43 64 80)(11 19 44 49 65)(12 20 45 50 66)(13 21 46 51 67)(14 22 47 52 68)(15 23 48 53 69)(16 24 33 54 70)
(1 79)(2 80)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(33 35)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(62 64)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(78 80)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111122222222444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8D8D10D10D16D4xD5D5xD8D5xD16
kernelD5xD16D5xC16D80C5:D16C5xD16D5xD8C5:2C8C4xD5D16Dic5D10C16D8D5C4C2C1
# reps11121211222248248

Matrix representation of D5xD16 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] >;

D5xD16 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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