metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2D16, D8⋊1D5, D40⋊3C2, C20.3D4, C10.8D8, C8.4D10, C40.2C22, (C5×D8)⋊1C2, C5⋊2C16⋊1C2, C2.4(D4⋊D5), C4.1(C5⋊D4), SmallGroup(160,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D16
G = < a,b,c | a5=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C5⋊D16
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 16A | 16B | 16C | 16D | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 8 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ7 | 2 | 2 | 2 | 0 | 2 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | -ζ167+ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | 0 | 0 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D16 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ165-ζ163 | -ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | 0 | 0 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D16 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ167-ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | 0 | 0 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D16 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | -ζ165+ζ163 | ζ167-ζ16 | -ζ167+ζ16 | ζ165-ζ163 | 0 | 0 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D16 |
| ρ16 | 2 | 2 | 0 | 0 | 2 | -1-√5/2 | -1+√5/2 | -2 | -2 | -1-√5/2 | -1+√5/2 | ζ53-ζ52 | ζ54-ζ5 | -ζ53+ζ52 | -ζ54+ζ5 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | -1+√5/2 | -1-√5/2 | -2 | -2 | -1+√5/2 | -1-√5/2 | ζ54-ζ5 | -ζ53+ζ52 | -ζ54+ζ5 | ζ53-ζ52 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ18 | 2 | 2 | 0 | 0 | 2 | -1-√5/2 | -1+√5/2 | -2 | -2 | -1-√5/2 | -1+√5/2 | -ζ53+ζ52 | -ζ54+ζ5 | ζ53-ζ52 | ζ54-ζ5 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ19 | 2 | 2 | 0 | 0 | 2 | -1+√5/2 | -1-√5/2 | -2 | -2 | -1+√5/2 | -1-√5/2 | -ζ54+ζ5 | ζ53-ζ52 | ζ54-ζ5 | -ζ53+ζ52 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ20 | 4 | 4 | 0 | 0 | -4 | -1-√5 | -1+√5 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊D5, Schur index 2 |
| ρ21 | 4 | 4 | 0 | 0 | -4 | -1+√5 | -1-√5 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊D5, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2√2 | 2√2 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 | ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | orthogonal faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2√2 | -2√2 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 | ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | orthogonal faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2√2 | 2√2 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 | ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 | orthogonal faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2√2 | -2√2 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 | ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 | orthogonal faithful, Schur index 2 |
►On 80 points
Generators in S80
(1 63 27 44 73)(2 74 45 28 64)(3 49 29 46 75)(4 76 47 30 50)(5 51 31 48 77)(6 78 33 32 52)(7 53 17 34 79)(8 80 35 18 54)(9 55 19 36 65)(10 66 37 20 56)(11 57 21 38 67)(12 68 39 22 58)(13 59 23 40 69)(14 70 41 24 60)(15 61 25 42 71)(16 72 43 26 62)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)
G:=sub<Sym(80)| (1,63,27,44,73)(2,74,45,28,64)(3,49,29,46,75)(4,76,47,30,50)(5,51,31,48,77)(6,78,33,32,52)(7,53,17,34,79)(8,80,35,18,54)(9,55,19,36,65)(10,66,37,20,56)(11,57,21,38,67)(12,68,39,22,58)(13,59,23,40,69)(14,70,41,24,60)(15,61,25,42,71)(16,72,43,26,62), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)>;
G:=Group( (1,63,27,44,73)(2,74,45,28,64)(3,49,29,46,75)(4,76,47,30,50)(5,51,31,48,77)(6,78,33,32,52)(7,53,17,34,79)(8,80,35,18,54)(9,55,19,36,65)(10,66,37,20,56)(11,57,21,38,67)(12,68,39,22,58)(13,59,23,40,69)(14,70,41,24,60)(15,61,25,42,71)(16,72,43,26,62), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72) );
G=PermutationGroup([[(1,63,27,44,73),(2,74,45,28,64),(3,49,29,46,75),(4,76,47,30,50),(5,51,31,48,77),(6,78,33,32,52),(7,53,17,34,79),(8,80,35,18,54),(9,55,19,36,65),(10,66,37,20,56),(11,57,21,38,67),(12,68,39,22,58),(13,59,23,40,69),(14,70,41,24,60),(15,61,25,42,71),(16,72,43,26,62)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,80),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72)]])
C5⋊D16 is a maximal subgroup of
D5×D16 D16⋊D5 C16⋊D10 SD32⋊3D5 D8.D10 D8⋊D10 C40.30C23 C15⋊D16 C5⋊D48 C15⋊7D16
C5⋊D16 is a maximal quotient of
C40.2Q8 C40.5D4 C5⋊D32 D16.D5 C5⋊SD64 C5⋊Q64 C10.D16 C15⋊D16 C5⋊D48 C15⋊7D16
Matrix representation of C5⋊D16 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 50 | 190 |
| 58 | 54 | 0 | 0 |
| 214 | 112 | 0 | 0 |
| 0 | 0 | 69 | 27 |
| 0 | 0 | 20 | 172 |
| 1 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 |
| 0 | 0 | 51 | 1 |
| 0 | 0 | 51 | 190 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[58,214,0,0,54,112,0,0,0,0,69,20,0,0,27,172],[1,1,0,0,0,240,0,0,0,0,51,51,0,0,1,190] >;
C5⋊D16 in GAP, Magma, Sage, TeX
C_5\rtimes D_{16} % in TeX
G:=Group("C5:D16"); // GroupNames label
G:=SmallGroup(160,33);
// by ID
G=gap.SmallGroup(160,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,218,116,122,579,297,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^5=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C5⋊D16 in TeX
Character table of C5⋊D16 in TeX