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
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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