metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5⋊C16, C8.5F5, C40.4C4, D10.2C8, Dic5.2C8, C5⋊C16⋊3C2, C5⋊1(C2×C16), C10.1(C2×C8), (C4×D5).5C4, (C8×D5).7C2, C4.14(C2×F5), C20.13(C2×C4), C2.1(D5⋊C8), C5⋊2C8.14C22, SmallGroup(160,64)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5⋊C16 |
Generators and relations for D5⋊C16
G = < a,b,c | a5=b2=c16=1, bab=a-1, cac-1=a3, cbc-1=a2b >
Smallest permutation representation of D5⋊C16
►On 80 points
Generators in S80
(1 62 68 47 25)(2 48 63 26 69)(3 27 33 70 64)(4 71 28 49 34)(5 50 72 35 29)(6 36 51 30 73)(7 31 37 74 52)(8 75 32 53 38)(9 54 76 39 17)(10 40 55 18 77)(11 19 41 78 56)(12 79 20 57 42)(13 58 80 43 21)(14 44 59 22 65)(15 23 45 66 60)(16 67 24 61 46)
(1 25)(2 69)(3 64)(4 34)(5 29)(6 73)(7 52)(8 38)(9 17)(10 77)(11 56)(12 42)(13 21)(14 65)(15 60)(16 46)(18 40)(19 78)(22 44)(23 66)(26 48)(27 70)(30 36)(31 74)(35 50)(39 54)(43 58)(47 62)(49 71)(53 75)(57 79)(61 67)
(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)
G:=sub<Sym(80)| (1,62,68,47,25)(2,48,63,26,69)(3,27,33,70,64)(4,71,28,49,34)(5,50,72,35,29)(6,36,51,30,73)(7,31,37,74,52)(8,75,32,53,38)(9,54,76,39,17)(10,40,55,18,77)(11,19,41,78,56)(12,79,20,57,42)(13,58,80,43,21)(14,44,59,22,65)(15,23,45,66,60)(16,67,24,61,46), (1,25)(2,69)(3,64)(4,34)(5,29)(6,73)(7,52)(8,38)(9,17)(10,77)(11,56)(12,42)(13,21)(14,65)(15,60)(16,46)(18,40)(19,78)(22,44)(23,66)(26,48)(27,70)(30,36)(31,74)(35,50)(39,54)(43,58)(47,62)(49,71)(53,75)(57,79)(61,67), (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)>;
G:=Group( (1,62,68,47,25)(2,48,63,26,69)(3,27,33,70,64)(4,71,28,49,34)(5,50,72,35,29)(6,36,51,30,73)(7,31,37,74,52)(8,75,32,53,38)(9,54,76,39,17)(10,40,55,18,77)(11,19,41,78,56)(12,79,20,57,42)(13,58,80,43,21)(14,44,59,22,65)(15,23,45,66,60)(16,67,24,61,46), (1,25)(2,69)(3,64)(4,34)(5,29)(6,73)(7,52)(8,38)(9,17)(10,77)(11,56)(12,42)(13,21)(14,65)(15,60)(16,46)(18,40)(19,78)(22,44)(23,66)(26,48)(27,70)(30,36)(31,74)(35,50)(39,54)(43,58)(47,62)(49,71)(53,75)(57,79)(61,67), (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) );
G=PermutationGroup([[(1,62,68,47,25),(2,48,63,26,69),(3,27,33,70,64),(4,71,28,49,34),(5,50,72,35,29),(6,36,51,30,73),(7,31,37,74,52),(8,75,32,53,38),(9,54,76,39,17),(10,40,55,18,77),(11,19,41,78,56),(12,79,20,57,42),(13,58,80,43,21),(14,44,59,22,65),(15,23,45,66,60),(16,67,24,61,46)], [(1,25),(2,69),(3,64),(4,34),(5,29),(6,73),(7,52),(8,38),(9,17),(10,77),(11,56),(12,42),(13,21),(14,65),(15,60),(16,46),(18,40),(19,78),(22,44),(23,66),(26,48),(27,70),(30,36),(31,74),(35,50),(39,54),(43,58),(47,62),(49,71),(53,75),(57,79),(61,67)], [(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)]])
D5⋊C16 is a maximal subgroup of
C16×F5 C16⋊7F5 D5.D16 D8.F5 D5.Q32 Q16.F5 D5⋊M5(2) Dic10.C8 D15⋊C16 C24.F5
D5⋊C16 is a maximal quotient of
D5⋊C32 C80.C4 Dic5⋊C16 D10⋊C16 C10.M5(2) D15⋊C16 C24.F5
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 16A | ··· | 16P | 20A | 20B | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | ··· | 16 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 4 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | F5 | C2×F5 | D5⋊C8 | D5⋊C16 |
| kernel | D5⋊C16 | C5⋊C16 | C8×D5 | C40 | C4×D5 | Dic5 | D10 | D5 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of D5⋊C16 ►in GL4(𝔽241) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 240 | 240 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 229 | 229 |
| 229 | 229 | 0 | 40 |
| 12 | 52 | 12 | 0 |
| 201 | 189 | 189 | 201 |
G:=sub<GL(4,GF(241))| [0,0,0,240,1,0,0,240,0,1,0,240,0,0,1,240],[0,1,240,0,1,0,240,0,0,0,240,0,0,0,240,1],[40,229,12,201,0,229,52,189,229,0,12,189,229,40,0,201] >;
D5⋊C16 in GAP, Magma, Sage, TeX
D_5\rtimes C_{16} % in TeX
G:=Group("D5:C16"); // GroupNames label
G:=SmallGroup(160,64);
// by ID
G=gap.SmallGroup(160,64);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,50,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c|a^5=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b>;
// generators/relations
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