direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C16, C80⋊4C2, D10.4C8, C8.19D10, Dic5.4C8, C40.19C22, C5⋊3(C2×C16), C5⋊2C16⋊6C2, C2.1(C8×D5), C5⋊2C8.7C4, (C4×D5).9C4, C4.16(C4×D5), C20.42(C2×C4), C10.10(C2×C8), (C8×D5).11C2, SmallGroup(160,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C16 |
Generators and relations for D5×C16
G = < a,b,c | a16=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C16
►On 80 points
Generators in S80
(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 20 42 60 68)(2 21 43 61 69)(3 22 44 62 70)(4 23 45 63 71)(5 24 46 64 72)(6 25 47 49 73)(7 26 48 50 74)(8 27 33 51 75)(9 28 34 52 76)(10 29 35 53 77)(11 30 36 54 78)(12 31 37 55 79)(13 32 38 56 80)(14 17 39 57 65)(15 18 40 58 66)(16 19 41 59 67)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
G:=sub<Sym(80)| (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), (1,20,42,60,68)(2,21,43,61,69)(3,22,44,62,70)(4,23,45,63,71)(5,24,46,64,72)(6,25,47,49,73)(7,26,48,50,74)(8,27,33,51,75)(9,28,34,52,76)(10,29,35,53,77)(11,30,36,54,78)(12,31,37,55,79)(13,32,38,56,80)(14,17,39,57,65)(15,18,40,58,66)(16,19,41,59,67), (1,76)(2,77)(3,78)(4,79)(5,80)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)>;
G:=Group( (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), (1,20,42,60,68)(2,21,43,61,69)(3,22,44,62,70)(4,23,45,63,71)(5,24,46,64,72)(6,25,47,49,73)(7,26,48,50,74)(8,27,33,51,75)(9,28,34,52,76)(10,29,35,53,77)(11,30,36,54,78)(12,31,37,55,79)(13,32,38,56,80)(14,17,39,57,65)(15,18,40,58,66)(16,19,41,59,67), (1,76)(2,77)(3,78)(4,79)(5,80)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48) );
G=PermutationGroup([[(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)], [(1,20,42,60,68),(2,21,43,61,69),(3,22,44,62,70),(4,23,45,63,71),(5,24,46,64,72),(6,25,47,49,73),(7,26,48,50,74),(8,27,33,51,75),(9,28,34,52,76),(10,29,35,53,77),(11,30,36,54,78),(12,31,37,55,79),(13,32,38,56,80),(14,17,39,57,65),(15,18,40,58,66),(16,19,41,59,67)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)]])
D5×C16 is a maximal subgroup of
C32⋊D5 D5⋊C32 C80.C4 C16⋊7F5 C80⋊2C4 C80⋊3C4 C16.F5 C80.2C4 D20.6C8 D20.5C8 D16⋊3D5 SD32⋊3D5 D80⋊5C2 D15⋊2C16
D5×C16 is a maximal quotient of
C32⋊D5 C40.88D4 D10⋊1C16 D15⋊2C16
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 20C | 20D | 40A | ··· | 40H | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D5 | D10 | C4×D5 | C8×D5 | D5×C16 |
| kernel | D5×C16 | C5⋊2C16 | C80 | C8×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | D5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 4 | 8 | 16 |
Matrix representation of D5×C16 ►in GL2(𝔽241) generated by
| 111 | 0 |
| 0 | 111 |
| 189 | 1 |
| 240 | 0 |
| 0 | 240 |
| 240 | 0 |
G:=sub<GL(2,GF(241))| [111,0,0,111],[189,240,1,0],[0,240,240,0] >;
D5×C16 in GAP, Magma, Sage, TeX
D_5\times C_{16} % in TeX
G:=Group("D5xC16"); // GroupNames label
G:=SmallGroup(160,4);
// by ID
G=gap.SmallGroup(160,4);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,31,50,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^16=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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