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## G = C5×Q16order 80 = 24·5

### Direct product of C5 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×Q16, C8.C10, Q8.C10, C40.3C2, C10.16D4, C20.19C22, C2.5(C5×D4), C4.3(C2×C10), (C5×Q8).2C2, SmallGroup(80,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×Q16
 Chief series C1 — C2 — C4 — C20 — C5×Q8 — C5×Q16
 Lower central C1 — C2 — C4 — C5×Q16
 Upper central C1 — C10 — C20 — C5×Q16

Generators and relations for C5×Q16
G = < a,b,c | a5=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Q16
Regular action on 80 points
Generators in S80
(1 58 21 75 30)(2 59 22 76 31)(3 60 23 77 32)(4 61 24 78 25)(5 62 17 79 26)(6 63 18 80 27)(7 64 19 73 28)(8 57 20 74 29)(9 48 56 69 35)(10 41 49 70 36)(11 42 50 71 37)(12 43 51 72 38)(13 44 52 65 39)(14 45 53 66 40)(15 46 54 67 33)(16 47 55 68 34)
(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 42 5 46)(2 41 6 45)(3 48 7 44)(4 47 8 43)(9 28 13 32)(10 27 14 31)(11 26 15 30)(12 25 16 29)(17 67 21 71)(18 66 22 70)(19 65 23 69)(20 72 24 68)(33 75 37 79)(34 74 38 78)(35 73 39 77)(36 80 40 76)(49 63 53 59)(50 62 54 58)(51 61 55 57)(52 60 56 64)

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

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

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

C5×Q16 is a maximal subgroup of   C5⋊SD32  C5⋊Q32  Q16⋊D5  Q8.D10

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C5 C10 C10 D4 Q16 C5×D4 C5×Q16 kernel C5×Q16 C40 C5×Q8 Q16 C8 Q8 C10 C5 C2 C1 # reps 1 1 2 4 4 8 1 2 4 8

Matrix representation of C5×Q16 in GL2(𝔽31) generated by

 8 0 0 8
,
 14 30 23 25
,
 0 29 16 0
G:=sub<GL(2,GF(31))| [8,0,0,8],[14,23,30,25],[0,16,29,0] >;

C5×Q16 in GAP, Magma, Sage, TeX

C_5\times Q_{16}
% in TeX

G:=Group("C5xQ16");
// GroupNames label

G:=SmallGroup(80,27);
// by ID

G=gap.SmallGroup(80,27);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,200,221,206,1203,608,58]);
// Polycyclic

G:=Group<a,b,c|a^5=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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