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 [×4], C4, C4 [×2], C22 [×6], C5, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, C16, C16, C2×C8, D8, D8 [×3], SD16, Q16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, M5(2), D16 [×2], SD32, SD32, C2×D8, C4○D8, C5⋊2C8, C40, C4×D5, C4×D5, D20 [×3], C5⋊D4, C5×D4, C5×Q8, C22×D5, C16⋊C22, C5⋊2C16, C80, C8×D5, D40 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], 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 31 70 36 58)(2 22 71 43 59 8 32 77 37 49)(3 29 72 34 60 15 17 68 38 56)(4 20 73 41 61 6 18 75 39 63)(5 27 74 48 62 13 19 66 40 54)(7 25 76 46 64 11 21 80 42 52)(9 23 78 44 50)(10 30 79 35 51 16 24 69 45 57)(12 28 65 33 53 14 26 67 47 55)
(1 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(17 34)(18 33)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(65 75)(66 74)(67 73)(68 72)(69 71)(76 80)(77 79)
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), (1,31,70,36,58)(2,22,71,43,59,8,32,77,37,49)(3,29,72,34,60,15,17,68,38,56)(4,20,73,41,61,6,18,75,39,63)(5,27,74,48,62,13,19,66,40,54)(7,25,76,46,64,11,21,80,42,52)(9,23,78,44,50)(10,30,79,35,51,16,24,69,45,57)(12,28,65,33,53,14,26,67,47,55), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,34)(18,33)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(65,75)(66,74)(67,73)(68,72)(69,71)(76,80)(77,79)>;
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), (1,31,70,36,58)(2,22,71,43,59,8,32,77,37,49)(3,29,72,34,60,15,17,68,38,56)(4,20,73,41,61,6,18,75,39,63)(5,27,74,48,62,13,19,66,40,54)(7,25,76,46,64,11,21,80,42,52)(9,23,78,44,50)(10,30,79,35,51,16,24,69,45,57)(12,28,65,33,53,14,26,67,47,55), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,34)(18,33)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(65,75)(66,74)(67,73)(68,72)(69,71)(76,80)(77,79) );
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)], [(1,31,70,36,58),(2,22,71,43,59,8,32,77,37,49),(3,29,72,34,60,15,17,68,38,56),(4,20,73,41,61,6,18,75,39,63),(5,27,74,48,62,13,19,66,40,54),(7,25,76,46,64,11,21,80,42,52),(9,23,78,44,50),(10,30,79,35,51,16,24,69,45,57),(12,28,65,33,53,14,26,67,47,55)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,50),(10,49),(11,64),(12,63),(13,62),(14,61),(15,60),(16,59),(17,34),(18,33),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(65,75),(66,74),(67,73),(68,72),(69,71),(76,80),(77,79)])
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