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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

 Derived series C1 — C8 — C5×D16
 Chief series C1 — C2 — C4 — C8 — C40 — C5×D8 — C5×D16
 Lower central C1 — C2 — C4 — C8 — C5×D16
 Upper central C1 — C10 — C20 — C40 — C5×D16

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

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 2A 2B 2C 4 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E ··· 10L 16A 16B 16C 16D 20A 20B 20C 20D 40A ··· 40H 80A ··· 80P order 1 2 2 2 4 5 5 5 5 8 8 10 10 10 10 10 ··· 10 16 16 16 16 20 20 20 20 40 ··· 40 80 ··· 80 size 1 1 8 8 2 1 1 1 1 2 2 1 1 1 1 8 ··· 8 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C5 C10 C10 D4 D8 D16 C5×D4 C5×D8 C5×D16 kernel C5×D16 C80 C5×D8 D16 C16 D8 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 8 1 2 4 4 8 16

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

 2 0 0 2
,
 0 23 4 14
,
 17 21 4 14
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

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