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G = C52C16order 80 = 24·5

The semidirect product of C5 and C16 acting via C16/C8=C2

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

Aliases: C52C16, C8.2D5, C10.2C8, C40.2C2, C20.5C4, C4.2Dic5, C2.(C52C8), SmallGroup(80,1)

Series: Derived Chief Lower central Upper central

C1C5 — C52C16
C1C5C10C20C40 — C52C16
C5 — C52C16
C1C8

Generators and relations for C52C16
 G = < a,b | a5=b16=1, bab-1=a-1 >

5C16

Smallest permutation representation of C52C16
Regular action on 80 points
Generators in S80
(1 76 24 54 45)(2 46 55 25 77)(3 78 26 56 47)(4 48 57 27 79)(5 80 28 58 33)(6 34 59 29 65)(7 66 30 60 35)(8 36 61 31 67)(9 68 32 62 37)(10 38 63 17 69)(11 70 18 64 39)(12 40 49 19 71)(13 72 20 50 41)(14 42 51 21 73)(15 74 22 52 43)(16 44 53 23 75)
(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)| (1,76,24,54,45)(2,46,55,25,77)(3,78,26,56,47)(4,48,57,27,79)(5,80,28,58,33)(6,34,59,29,65)(7,66,30,60,35)(8,36,61,31,67)(9,68,32,62,37)(10,38,63,17,69)(11,70,18,64,39)(12,40,49,19,71)(13,72,20,50,41)(14,42,51,21,73)(15,74,22,52,43)(16,44,53,23,75), (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( (1,76,24,54,45)(2,46,55,25,77)(3,78,26,56,47)(4,48,57,27,79)(5,80,28,58,33)(6,34,59,29,65)(7,66,30,60,35)(8,36,61,31,67)(9,68,32,62,37)(10,38,63,17,69)(11,70,18,64,39)(12,40,49,19,71)(13,72,20,50,41)(14,42,51,21,73)(15,74,22,52,43)(16,44,53,23,75), (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([(1,76,24,54,45),(2,46,55,25,77),(3,78,26,56,47),(4,48,57,27,79),(5,80,28,58,33),(6,34,59,29,65),(7,66,30,60,35),(8,36,61,31,67),(9,68,32,62,37),(10,38,63,17,69),(11,70,18,64,39),(12,40,49,19,71),(13,72,20,50,41),(14,42,51,21,73),(15,74,22,52,43),(16,44,53,23,75)], [(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)])

C52C16 is a maximal subgroup of
C5⋊C32  D5×C16  C80⋊C2  C20.4C8  C5⋊D16  D8.D5  C5⋊SD32  C5⋊Q32  C153C16  C252C16  C527C16  C523C16
C52C16 is a maximal quotient of
C52C32  C153C16  C252C16  C527C16  C523C16

32 conjugacy classes

class 1  2 4A4B5A5B8A8B8C8D10A10B16A···16H20A20B20C20D40A···40H
order1244558888101016···162020202040···40
size1111221111225···522222···2

32 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16D5Dic5C52C8C52C16
kernelC52C16C40C20C10C5C8C4C2C1
# reps112482248

Matrix representation of C52C16 in GL2(𝔽41) generated by

4018
3935
,
2923
1512
G:=sub<GL(2,GF(41))| [40,39,18,35],[29,15,23,12] >;

C52C16 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_{16}
% in TeX

G:=Group("C5:2C16");
// GroupNames label

G:=SmallGroup(80,1);
// by ID

G=gap.SmallGroup(80,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,10,26,42,1604]);
// Polycyclic

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

Export

Subgroup lattice of C52C16 in TeX

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