metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5⋊C16, C10.C8, C4.2F5, C20.2C4, C2.(C5⋊C8), C5⋊2C8.2C2, SmallGroup(80,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5⋊C16 |
Generators and relations for C5⋊C16
G = < a,b | a5=b16=1, bab-1=a3 >
Character table of C5⋊C16
| class | 1 | 2 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | |
| size | 1 | 1 | 1 | 1 | 4 | 5 | 5 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | i | -i | -i | -i | i | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | -i | i | i | i | -i | 1 | 1 | linear of order 4 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | -1 | -1 | linear of order 8 |
| ρ9 | 1 | -1 | i | -i | 1 | ζ1614 | ζ162 | ζ1610 | ζ166 | -1 | ζ1611 | ζ1613 | ζ169 | ζ16 | ζ1615 | ζ167 | ζ163 | ζ165 | -i | i | linear of order 16 |
| ρ10 | 1 | -1 | i | -i | 1 | ζ1614 | ζ162 | ζ1610 | ζ166 | -1 | ζ163 | ζ165 | ζ16 | ζ169 | ζ167 | ζ1615 | ζ1611 | ζ1613 | -i | i | linear of order 16 |
| ρ11 | 1 | -1 | -i | i | 1 | ζ1610 | ζ166 | ζ1614 | ζ162 | -1 | ζ169 | ζ1615 | ζ163 | ζ1611 | ζ165 | ζ1613 | ζ16 | ζ167 | i | -i | linear of order 16 |
| ρ12 | 1 | -1 | -i | i | 1 | ζ1610 | ζ166 | ζ1614 | ζ162 | -1 | ζ16 | ζ167 | ζ1611 | ζ163 | ζ1613 | ζ165 | ζ169 | ζ1615 | i | -i | linear of order 16 |
| ρ13 | 1 | -1 | i | -i | 1 | ζ166 | ζ1610 | ζ162 | ζ1614 | -1 | ζ1615 | ζ169 | ζ165 | ζ1613 | ζ163 | ζ1611 | ζ167 | ζ16 | -i | i | linear of order 16 |
| ρ14 | 1 | -1 | i | -i | 1 | ζ166 | ζ1610 | ζ162 | ζ1614 | -1 | ζ167 | ζ16 | ζ1613 | ζ165 | ζ1611 | ζ163 | ζ1615 | ζ169 | -i | i | linear of order 16 |
| ρ15 | 1 | -1 | -i | i | 1 | ζ162 | ζ1614 | ζ166 | ζ1610 | -1 | ζ1613 | ζ1611 | ζ1615 | ζ167 | ζ169 | ζ16 | ζ165 | ζ163 | i | -i | linear of order 16 |
| ρ16 | 1 | -1 | -i | i | 1 | ζ162 | ζ1614 | ζ166 | ζ1610 | -1 | ζ165 | ζ163 | ζ167 | ζ1615 | ζ16 | ζ169 | ζ1613 | ζ1611 | i | -i | linear of order 16 |
| ρ17 | 4 | 4 | 4 | 4 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ18 | 4 | 4 | -4 | -4 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ19 | 4 | -4 | 4i | -4i | -1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | complex faithful, Schur index 4 |
| ρ20 | 4 | -4 | -4i | 4i | -1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | complex faithful, Schur index 4 |
►Regular action on 80 points
Generators in S80
(1 72 25 45 59)(2 46 73 60 26)(3 61 47 27 74)(4 28 62 75 48)(5 76 29 33 63)(6 34 77 64 30)(7 49 35 31 78)(8 32 50 79 36)(9 80 17 37 51)(10 38 65 52 18)(11 53 39 19 66)(12 20 54 67 40)(13 68 21 41 55)(14 42 69 56 22)(15 57 43 23 70)(16 24 58 71 44)
(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)| (1,72,25,45,59)(2,46,73,60,26)(3,61,47,27,74)(4,28,62,75,48)(5,76,29,33,63)(6,34,77,64,30)(7,49,35,31,78)(8,32,50,79,36)(9,80,17,37,51)(10,38,65,52,18)(11,53,39,19,66)(12,20,54,67,40)(13,68,21,41,55)(14,42,69,56,22)(15,57,43,23,70)(16,24,58,71,44), (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( (1,72,25,45,59)(2,46,73,60,26)(3,61,47,27,74)(4,28,62,75,48)(5,76,29,33,63)(6,34,77,64,30)(7,49,35,31,78)(8,32,50,79,36)(9,80,17,37,51)(10,38,65,52,18)(11,53,39,19,66)(12,20,54,67,40)(13,68,21,41,55)(14,42,69,56,22)(15,57,43,23,70)(16,24,58,71,44), (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([[(1,72,25,45,59),(2,46,73,60,26),(3,61,47,27,74),(4,28,62,75,48),(5,76,29,33,63),(6,34,77,64,30),(7,49,35,31,78),(8,32,50,79,36),(9,80,17,37,51),(10,38,65,52,18),(11,53,39,19,66),(12,20,54,67,40),(13,68,21,41,55),(14,42,69,56,22),(15,57,43,23,70),(16,24,58,71,44)], [(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)]])
C5⋊C16 is a maximal subgroup of
D5⋊C16 C8.F5 C20.C8 C15⋊C16 C25⋊C16 C52⋊3C16 C52⋊4C16 C52⋊5C16
C5⋊C16 is a maximal quotient of
C5⋊C32 C15⋊C16 C25⋊C16 C52⋊3C16 C52⋊4C16 C52⋊5C16
Matrix representation of C5⋊C16 ►in GL4(𝔽241) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 240 | 240 | 240 |
| 183 | 221 | 133 | 90 |
| 153 | 110 | 20 | 203 |
| 151 | 93 | 131 | 43 |
| 38 | 191 | 148 | 58 |
G:=sub<GL(4,GF(241))| [0,0,0,240,1,0,0,240,0,1,0,240,0,0,1,240],[183,153,151,38,221,110,93,191,133,20,131,148,90,203,43,58] >;
C5⋊C16 in GAP, Magma, Sage, TeX
C_5\rtimes C_{16} % in TeX
G:=Group("C5:C16"); // GroupNames label
G:=SmallGroup(80,3);
// by ID
G=gap.SmallGroup(80,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,10,26,42,804,809]);
// Polycyclic
G:=Group<a,b|a^5=b^16=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C5⋊C16 in TeX
Character table of C5⋊C16 in TeX