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G = D5⋊C16order 160 = 25·5

The semidirect product of D5 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5⋊C16, C8.5F5, C40.4C4, D10.2C8, Dic5.2C8, C5⋊C163C2, C51(C2×C16), C10.1(C2×C8), (C4×D5).5C4, (C8×D5).7C2, C4.14(C2×F5), C20.13(C2×C4), C2.1(D5⋊C8), C52C8.14C22, SmallGroup(160,64)

Series: Derived Chief Lower central Upper central

C1C5 — D5⋊C16
C1C5C10C20C52C8C5⋊C16 — D5⋊C16
C5 — D5⋊C16
C1C8

Generators and relations for D5⋊C16
 G = < a,b,c | a5=b2=c16=1, bab=a-1, cac-1=a3, cbc-1=a2b >

5C2
5C2
5C22
5C4
5C8
5C2×C4
5C16
5C2×C8
5C16
5C2×C16

Smallest permutation representation of D5⋊C16
On 80 points
Generators in S80
(1 62 68 47 25)(2 48 63 26 69)(3 27 33 70 64)(4 71 28 49 34)(5 50 72 35 29)(6 36 51 30 73)(7 31 37 74 52)(8 75 32 53 38)(9 54 76 39 17)(10 40 55 18 77)(11 19 41 78 56)(12 79 20 57 42)(13 58 80 43 21)(14 44 59 22 65)(15 23 45 66 60)(16 67 24 61 46)
(1 25)(2 69)(3 64)(4 34)(5 29)(6 73)(7 52)(8 38)(9 17)(10 77)(11 56)(12 42)(13 21)(14 65)(15 60)(16 46)(18 40)(19 78)(22 44)(23 66)(26 48)(27 70)(30 36)(31 74)(35 50)(39 54)(43 58)(47 62)(49 71)(53 75)(57 79)(61 67)
(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)

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

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

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

D5⋊C16 is a maximal subgroup of
C16×F5  C167F5  D5.D16  D8.F5  D5.Q32  Q16.F5  D5⋊M5(2)  Dic10.C8  D15⋊C16  C24.F5
D5⋊C16 is a maximal quotient of
D5⋊C32  C80.C4  Dic5⋊C16  D10⋊C16  C10.M5(2)  D15⋊C16  C24.F5

40 conjugacy classes

class 1 2A2B2C4A4B4C4D 5 8A8B8C8D8E8F8G8H 10 16A···16P20A20B40A40B40C40D
order122244445888888881016···16202040404040
size1155115541111555545···5444444

40 irreducible representations

dim111111114444
type+++++
imageC1C2C2C4C4C8C8C16F5C2×F5D5⋊C8D5⋊C16
kernelD5⋊C16C5⋊C16C8×D5C40C4×D5Dic5D10D5C8C4C2C1
# reps1212244161124

Matrix representation of D5⋊C16 in GL4(𝔽241) generated by

0100
0010
0001
240240240240
,
0100
1000
240240240240
0001
,
400229229
229229040
1252120
201189189201
G:=sub<GL(4,GF(241))| [0,0,0,240,1,0,0,240,0,1,0,240,0,0,1,240],[0,1,240,0,1,0,240,0,0,0,240,0,0,0,240,1],[40,229,12,201,0,229,52,189,229,0,12,189,229,40,0,201] >;

D5⋊C16 in GAP, Magma, Sage, TeX

D_5\rtimes C_{16}
% in TeX

G:=Group("D5:C16");
// GroupNames label

G:=SmallGroup(160,64);
// by ID

G=gap.SmallGroup(160,64);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,50,69,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^5=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b>;
// generators/relations

Export

Subgroup lattice of D5⋊C16 in TeX

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