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G = C16⋊D5order 160 = 25·5

2nd semidirect product of C16 and D5 acting via D5/C5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C802C2, C162D5, C51SD32, C10.2D8, C2.4D40, C4.2D20, D40.1C2, C20.25D4, C8.14D10, Dic201C2, C40.15C22, SmallGroup(160,7)

Series: Derived Chief Lower central Upper central

C1C40 — C16⋊D5
C1C5C10C20C40D40 — C16⋊D5
C5C10C20C40 — C16⋊D5
C1C2C4C8C16

Generators and relations for C16⋊D5
 G = < a,b,c | a16=b5=c2=1, ab=ba, cac=a7, cbc=b-1 >

40C2
20C4
20C22
8D5
10Q8
10D4
4D10
4Dic5
5Q16
5D8
2Dic10
2D20
5SD32

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

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

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

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

C16⋊D5 is a maximal subgroup of
D807C2  D80⋊C2  C16.D10  D16⋊D5  D5×SD32  SD323D5  Q32⋊D5  D40.S3  C24.D10  C48⋊D5
C16⋊D5 is a maximal quotient of
C40.78D4  C8014C4  D407C4  D40.S3  C24.D10  C48⋊D5

43 conjugacy classes

class 1 2A2B4A4B5A5B8A8B10A10B16A16B16C16D20A20B20C20D40A···40H80A···80P
order1224455881010161616162020202040···4080···80
size1140240222222222222222···22···2

43 irreducible representations

dim111122222222
type++++++++++
imageC1C2C2C2D4D5D8D10SD32D20D40C16⋊D5
kernelC16⋊D5C80D40Dic20C20C16C10C8C5C4C2C1
# reps1111122244816

Matrix representation of C16⋊D5 in GL2(𝔽241) generated by

6147
162153
,
240191
240190
,
2400
2401
G:=sub<GL(2,GF(241))| [6,162,147,153],[240,240,191,190],[240,240,0,1] >;

C16⋊D5 in GAP, Magma, Sage, TeX

C_{16}\rtimes D_5
% in TeX

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

G:=SmallGroup(160,7);
// by ID

G=gap.SmallGroup(160,7);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,79,506,50,579,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^16=b^5=c^2=1,a*b=b*a,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C16⋊D5 in TeX

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