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G = C5×D16order 160 = 25·5

Direct product of C5 and D16

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×D16, C803C2, C161C10, D81C10, C10.15D8, C20.36D4, C40.24C22, (C5×D8)⋊5C2, C4.1(C5×D4), C2.3(C5×D8), C8.2(C2×C10), SmallGroup(160,61)

Series: Derived Chief Lower central Upper central

C1C8 — C5×D16
C1C2C4C8C40C5×D8 — C5×D16
C1C2C4C8 — C5×D16
C1C10C20C40 — C5×D16

Generators and relations for C5×D16
 G = < a,b,c | a5=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

8C2
8C2
4C22
4C22
8C10
8C10
2D4
2D4
4C2×C10
4C2×C10
2C5×D4
2C5×D4

Smallest permutation representation of C5×D16
On 80 points
Generators in S80
(1 45 59 67 18)(2 46 60 68 19)(3 47 61 69 20)(4 48 62 70 21)(5 33 63 71 22)(6 34 64 72 23)(7 35 49 73 24)(8 36 50 74 25)(9 37 51 75 26)(10 38 52 76 27)(11 39 53 77 28)(12 40 54 78 29)(13 41 55 79 30)(14 42 56 80 31)(15 43 57 65 32)(16 44 58 66 17)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 18)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 40)(34 39)(35 38)(36 37)(41 48)(42 47)(43 46)(44 45)(49 52)(50 51)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(65 68)(66 67)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)

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

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

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

C5×D16 is a maximal subgroup of   C5⋊D32  D16.D5  D16⋊D5  D163D5

55 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D8A8B10A10B10C10D10E···10L16A16B16C16D20A20B20C20D40A···40H80A···80P
order122245555881010101010···10161616162020202040···4080···80
size1188211112211118···8222222222···22···2

55 irreducible representations

dim111111222222
type++++++
imageC1C2C2C5C10C10D4D8D16C5×D4C5×D8C5×D16
kernelC5×D16C80C5×D8D16C16D8C20C10C5C4C2C1
# reps1124481244816

Matrix representation of C5×D16 in GL2(𝔽31) generated by

20
02
,
023
414
,
1721
414
G:=sub<GL(2,GF(31))| [2,0,0,2],[0,4,23,14],[17,4,21,14] >;

C5×D16 in GAP, Magma, Sage, TeX

C_5\times D_{16}
% in TeX

G:=Group("C5xD16");
// GroupNames label

G:=SmallGroup(160,61);
// by ID

G=gap.SmallGroup(160,61);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,265,1443,729,165,3604,1810,88]);
// Polycyclic

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

Export

Subgroup lattice of C5×D16 in TeX

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