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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

C1C4 — C5×Q16
C1C2C4C20C5×Q8 — C5×Q16
C1C2C4 — C5×Q16
C1C10C20 — 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 >

2C4
2C4
2C20
2C20

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 4A4B4C5A5B5C5D8A8B10A10B10C10D20A20B20C20D20E···20L40A···40H
order12444555588101010102020202020···2040···40
size11244111122111122224···42···2

35 irreducible representations

dim1111112222
type++++-
imageC1C2C2C5C10C10D4Q16C5×D4C5×Q16
kernelC5×Q16C40C5×Q8Q16C8Q8C10C5C2C1
# reps1124481248

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

80
08
,
1430
2325
,
029
160
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

Export

Subgroup lattice of C5×Q16 in TeX

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