direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×D16, C80⋊3C2, C16⋊1C10, D8⋊1C10, 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
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 29 63 38 80)(2 30 64 39 65)(3 31 49 40 66)(4 32 50 41 67)(5 17 51 42 68)(6 18 52 43 69)(7 19 53 44 70)(8 20 54 45 71)(9 21 55 46 72)(10 22 56 47 73)(11 23 57 48 74)(12 24 58 33 75)(13 25 59 34 76)(14 26 60 35 77)(15 27 61 36 78)(16 28 62 37 79)
(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 24)(18 23)(19 22)(20 21)(25 32)(26 31)(27 30)(28 29)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 64)(62 63)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)
G:=sub<Sym(80)| (1,29,63,38,80)(2,30,64,39,65)(3,31,49,40,66)(4,32,50,41,67)(5,17,51,42,68)(6,18,52,43,69)(7,19,53,44,70)(8,20,54,45,71)(9,21,55,46,72)(10,22,56,47,73)(11,23,57,48,74)(12,24,58,33,75)(13,25,59,34,76)(14,26,60,35,77)(15,27,61,36,78)(16,28,62,37,79), (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)(17,24)(18,23)(19,22)(20,21)(25,32)(26,31)(27,30)(28,29)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,64)(62,63)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)>;
G:=Group( (1,29,63,38,80)(2,30,64,39,65)(3,31,49,40,66)(4,32,50,41,67)(5,17,51,42,68)(6,18,52,43,69)(7,19,53,44,70)(8,20,54,45,71)(9,21,55,46,72)(10,22,56,47,73)(11,23,57,48,74)(12,24,58,33,75)(13,25,59,34,76)(14,26,60,35,77)(15,27,61,36,78)(16,28,62,37,79), (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)(17,24)(18,23)(19,22)(20,21)(25,32)(26,31)(27,30)(28,29)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,64)(62,63)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80) );
G=PermutationGroup([[(1,29,63,38,80),(2,30,64,39,65),(3,31,49,40,66),(4,32,50,41,67),(5,17,51,42,68),(6,18,52,43,69),(7,19,53,44,70),(8,20,54,45,71),(9,21,55,46,72),(10,22,56,47,73),(11,23,57,48,74),(12,24,58,33,75),(13,25,59,34,76),(14,26,60,35,77),(15,27,61,36,78),(16,28,62,37,79)], [(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),(17,24),(18,23),(19,22),(20,21),(25,32),(26,31),(27,30),(28,29),(33,42),(34,41),(35,40),(36,39),(37,38),(43,48),(44,47),(45,46),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,64),(62,63),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,80)]])
C5×D16 is a maximal subgroup of
C5⋊D32 D16.D5 D16⋊D5 D16⋊3D5
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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