metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2Q16, Q8.D5, C4.4D10, C10.10D4, C20.4C22, Dic10.2C2, C5⋊2C8.1C2, (C5×Q8).1C2, C2.7(C5⋊D4), SmallGroup(80,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊Q16
G = < a,b,c | a5=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Character table of C5⋊Q16
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 2 | 4 | 20 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | -2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -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 | 2 | -2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | 1+√5/2 | ζ54-ζ5 | complex lifted from C5⋊D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | 1+√5/2 | -ζ54+ζ5 | complex lifted from C5⋊D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | 1-√5/2 | ζ53-ζ52 | complex lifted from C5⋊D4 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | 1-√5/2 | -ζ53+ζ52 | complex lifted from C5⋊D4 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►Regular action on 80 points
Generators in S80
(1 66 77 34 10)(2 11 35 78 67)(3 68 79 36 12)(4 13 37 80 69)(5 70 73 38 14)(6 15 39 74 71)(7 72 75 40 16)(8 9 33 76 65)(17 29 60 50 42)(18 43 51 61 30)(19 31 62 52 44)(20 45 53 63 32)(21 25 64 54 46)(22 47 55 57 26)(23 27 58 56 48)(24 41 49 59 28)
(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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 38 21 34)(18 37 22 33)(19 36 23 40)(20 35 24 39)(41 74 45 78)(42 73 46 77)(43 80 47 76)(44 79 48 75)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
G:=sub<Sym(80)| (1,66,77,34,10)(2,11,35,78,67)(3,68,79,36,12)(4,13,37,80,69)(5,70,73,38,14)(6,15,39,74,71)(7,72,75,40,16)(8,9,33,76,65)(17,29,60,50,42)(18,43,51,61,30)(19,31,62,52,44)(20,45,53,63,32)(21,25,64,54,46)(22,47,55,57,26)(23,27,58,56,48)(24,41,49,59,28), (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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,30,13,26)(10,29,14,25)(11,28,15,32)(12,27,16,31)(17,38,21,34)(18,37,22,33)(19,36,23,40)(20,35,24,39)(41,74,45,78)(42,73,46,77)(43,80,47,76)(44,79,48,75)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72)>;
G:=Group( (1,66,77,34,10)(2,11,35,78,67)(3,68,79,36,12)(4,13,37,80,69)(5,70,73,38,14)(6,15,39,74,71)(7,72,75,40,16)(8,9,33,76,65)(17,29,60,50,42)(18,43,51,61,30)(19,31,62,52,44)(20,45,53,63,32)(21,25,64,54,46)(22,47,55,57,26)(23,27,58,56,48)(24,41,49,59,28), (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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,30,13,26)(10,29,14,25)(11,28,15,32)(12,27,16,31)(17,38,21,34)(18,37,22,33)(19,36,23,40)(20,35,24,39)(41,74,45,78)(42,73,46,77)(43,80,47,76)(44,79,48,75)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72) );
G=PermutationGroup([[(1,66,77,34,10),(2,11,35,78,67),(3,68,79,36,12),(4,13,37,80,69),(5,70,73,38,14),(6,15,39,74,71),(7,72,75,40,16),(8,9,33,76,65),(17,29,60,50,42),(18,43,51,61,30),(19,31,62,52,44),(20,45,53,63,32),(21,25,64,54,46),(22,47,55,57,26),(23,27,58,56,48),(24,41,49,59,28)], [(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)], [(1,60,5,64),(2,59,6,63),(3,58,7,62),(4,57,8,61),(9,30,13,26),(10,29,14,25),(11,28,15,32),(12,27,16,31),(17,38,21,34),(18,37,22,33),(19,36,23,40),(20,35,24,39),(41,74,45,78),(42,73,46,77),(43,80,47,76),(44,79,48,75),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)]])
C5⋊Q16 is a maximal subgroup of
SD16⋊D5 SD16⋊3D5 D5×Q16 Q16⋊D5 C20.C23 D4.8D10 D4.9D10 C15⋊Q16 C5⋊Dic12 C15⋊7Q16 Q8.D15 C25⋊Q16 C52⋊2Q16 C52⋊3Q16 C52⋊7Q16
C5⋊Q16 is a maximal quotient of
C10.D8 C10.Q16 Q8⋊Dic5 C15⋊Q16 C5⋊Dic12 C15⋊7Q16 C25⋊Q16 C52⋊2Q16 C52⋊3Q16 C52⋊7Q16
Matrix representation of C5⋊Q16 ►in GL4(𝔽41) generated by
| 6 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 1 | 0 | 0 |
| 6 | 35 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 27 |
| 0 | 0 | 27 | 7 |
G:=sub<GL(4,GF(41))| [6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[6,6,0,0,1,35,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,34,27,0,0,27,7] >;
C5⋊Q16 in GAP, Magma, Sage, TeX
C_5\rtimes Q_{16} % in TeX
G:=Group("C5:Q16"); // GroupNames label
G:=SmallGroup(80,18);
// by ID
G=gap.SmallGroup(80,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,40,61,46,182,97,42,1604]);
// Polycyclic
G:=Group<a,b,c|a^5=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C5⋊Q16 in TeX
Character table of C5⋊Q16 in TeX