metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D80⋊6C2, C16⋊3D10, C80⋊3C22, Q16⋊2D10, SD32⋊1D5, D8.3D10, D10.15D8, D40⋊6C22, C40.17C23, Dic5.17D8, (D5×D8)⋊5C2, C4.5(D4×D5), C5⋊D16⋊3C2, (C4×D5).8D4, C80⋊C2⋊1C2, C2.20(D5×D8), C5⋊2C8.3D4, C5⋊3(C16⋊C22), (C5×SD32)⋊1C2, C10.36(C2×D8), C20.11(C2×D4), C5⋊SD32⋊2C2, Q8.D10⋊3C2, C5⋊2C16⋊2C22, (C5×Q16)⋊5C22, (C8×D5).4C22, (C5×D8).3C22, C8.23(C22×D5), SmallGroup(320,541)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊D10
G = < a,b,c | a16=b10=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >
Subgroups: 566 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, M5(2), D16, SD32, SD32, C2×D8, C4○D8, C5⋊2C8, C40, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C16⋊C22, C5⋊2C16, C80, C8×D5, D40, D4⋊D5, Q8⋊D5, C5×D8, C5×Q16, D4×D5, Q8⋊2D5, C80⋊C2, D80, C5⋊D16, C5⋊SD32, C5×SD32, D5×D8, Q8.D10, C16⋊D10
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, C16⋊C22, D4×D5, D5×D8, C16⋊D10
►On 80 points
(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 53 36 22 73)(2 60 37 29 74 8 54 43 23 80)(3 51 38 20 75 15 55 34 24 71)(4 58 39 27 76 6 56 41 25 78)(5 49 40 18 77 13 57 48 26 69)(7 63 42 32 79 11 59 46 28 67)(9 61 44 30 65)(10 52 45 21 66 16 62 35 31 72)(12 50 47 19 68 14 64 33 17 70)
(1 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 64)(28 63)(29 62)(30 61)(31 60)(32 59)(33 39)(34 38)(35 37)(40 48)(41 47)(42 46)(43 45)
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), (1,53,36,22,73)(2,60,37,29,74,8,54,43,23,80)(3,51,38,20,75,15,55,34,24,71)(4,58,39,27,76,6,56,41,25,78)(5,49,40,18,77,13,57,48,26,69)(7,63,42,32,79,11,59,46,28,67)(9,61,44,30,65)(10,52,45,21,66,16,62,35,31,72)(12,50,47,19,68,14,64,33,17,70), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,39)(34,38)(35,37)(40,48)(41,47)(42,46)(43,45)>;
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), (1,53,36,22,73)(2,60,37,29,74,8,54,43,23,80)(3,51,38,20,75,15,55,34,24,71)(4,58,39,27,76,6,56,41,25,78)(5,49,40,18,77,13,57,48,26,69)(7,63,42,32,79,11,59,46,28,67)(9,61,44,30,65)(10,52,45,21,66,16,62,35,31,72)(12,50,47,19,68,14,64,33,17,70), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,39)(34,38)(35,37)(40,48)(41,47)(42,46)(43,45) );
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)], [(1,53,36,22,73),(2,60,37,29,74,8,54,43,23,80),(3,51,38,20,75,15,55,34,24,71),(4,58,39,27,76,6,56,41,25,78),(5,49,40,18,77,13,57,48,26,69),(7,63,42,32,79,11,59,46,28,67),(9,61,44,30,65),(10,52,45,21,66,16,62,35,31,72),(12,50,47,19,68,14,64,33,17,70)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,80),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,58),(18,57),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,64),(28,63),(29,62),(30,61),(31,60),(32,59),(33,39),(34,38),(35,37),(40,48),(41,47),(42,46),(43,45)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 10 | 40 | 40 | 2 | 8 | 10 | 2 | 2 | 2 | 2 | 20 | 2 | 2 | 16 | 16 | 4 | 4 | 20 | 20 | 4 | 4 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D10 | C16⋊C22 | D4×D5 | D5×D8 | C16⋊D10 |
| kernel | C16⋊D10 | C80⋊C2 | D80 | C5⋊D16 | C5⋊SD32 | C5×SD32 | D5×D8 | Q8.D10 | C5⋊2C8 | C4×D5 | SD32 | Dic5 | D10 | C16 | D8 | Q16 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C16⋊D10 ►in GL4(𝔽241) generated by
| 209 | 67 | 92 | 64 |
| 64 | 32 | 213 | 28 |
| 204 | 119 | 133 | 146 |
| 204 | 85 | 95 | 108 |
| 189 | 190 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 116 | 51 | 51 |
| 118 | 125 | 190 | 1 |
| 52 | 1 | 0 | 0 |
| 189 | 189 | 0 | 0 |
| 0 | 116 | 51 | 51 |
| 123 | 7 | 1 | 190 |
G:=sub<GL(4,GF(241))| [209,64,204,204,67,32,119,85,92,213,133,95,64,28,146,108],[189,52,0,118,190,0,116,125,0,0,51,190,0,0,51,1],[52,189,0,123,1,189,116,7,0,0,51,1,0,0,51,190] >;
C16⋊D10 in GAP, Magma, Sage, TeX
C_{16}\rtimes D_{10} % in TeX
G:=Group("C16:D10"); // GroupNames label
G:=SmallGroup(320,541);
// by ID
G=gap.SmallGroup(320,541);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,135,184,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^16=b^10=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations