metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D5, C40⋊4C2, C5⋊3M4(2), D10.1C4, C4.13D10, Dic5.1C4, C20.13C22, C5⋊2C8⋊4C2, C2.3(C4×D5), C10.9(C2×C4), (C4×D5).2C2, SmallGroup(80,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D5
G = < a,b,c | a8=b5=c2=1, ab=ba, cac=a5, cbc=b-1 >
Character table of C8⋊D5
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | 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 | -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 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | -2 | -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 | 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 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | -2 | -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 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ11 | 2 | 2 | 0 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 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 | -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 | 0 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 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 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ13 | 2 | -2 | 0 | -2i | 2i | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ14 | 2 | -2 | 0 | 2i | -2i | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ15 | 2 | 2 | 0 | -2 | -2 | 0 | -1+√5/2 | -1-√5/2 | -2i | 2i | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ53+ζ43ζ52 | complex lifted from C4×D5 |
| ρ16 | 2 | 2 | 0 | -2 | -2 | 0 | -1-√5/2 | -1+√5/2 | 2i | -2i | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ54+ζ4ζ5 | complex lifted from C4×D5 |
| ρ17 | 2 | 2 | 0 | -2 | -2 | 0 | -1+√5/2 | -1-√5/2 | 2i | -2i | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | complex lifted from C4×D5 |
| ρ18 | 2 | 2 | 0 | -2 | -2 | 0 | -1-√5/2 | -1+√5/2 | -2i | 2i | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | complex lifted from C4×D5 |
| ρ19 | 2 | -2 | 0 | -2i | 2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ86ζ53+ζ86ζ52 | ζ82ζ54+ζ82ζ5 | ζ86ζ54+ζ86ζ5 | ζ82ζ53+ζ82ζ52 | ζ83ζ54-ζ83ζ5 | ζ83ζ53-ζ83ζ52 | -ζ83ζ53+ζ83ζ52 | ζ87ζ54-ζ87ζ5 | ζ8ζ53-ζ8ζ52 | -ζ8ζ53+ζ8ζ52 | ζ85ζ54-ζ85ζ5 | ζ8ζ54-ζ8ζ5 | complex faithful |
| ρ20 | 2 | -2 | 0 | -2i | 2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ86ζ53+ζ86ζ52 | ζ82ζ54+ζ82ζ5 | ζ86ζ54+ζ86ζ5 | ζ82ζ53+ζ82ζ52 | ζ87ζ54-ζ87ζ5 | -ζ83ζ53+ζ83ζ52 | ζ83ζ53-ζ83ζ52 | ζ83ζ54-ζ83ζ5 | -ζ8ζ53+ζ8ζ52 | ζ8ζ53-ζ8ζ52 | ζ8ζ54-ζ8ζ5 | ζ85ζ54-ζ85ζ5 | complex faithful |
| ρ21 | 2 | -2 | 0 | -2i | 2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ86ζ54+ζ86ζ5 | ζ82ζ53+ζ82ζ52 | ζ86ζ53+ζ86ζ52 | ζ82ζ54+ζ82ζ5 | -ζ83ζ53+ζ83ζ52 | ζ83ζ54-ζ83ζ5 | ζ87ζ54-ζ87ζ5 | ζ83ζ53-ζ83ζ52 | ζ8ζ54-ζ8ζ5 | ζ85ζ54-ζ85ζ5 | ζ8ζ53-ζ8ζ52 | -ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ22 | 2 | -2 | 0 | 2i | -2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ82ζ54+ζ82ζ5 | ζ86ζ53+ζ86ζ52 | ζ82ζ53+ζ82ζ52 | ζ86ζ54+ζ86ζ5 | ζ8ζ53-ζ8ζ52 | ζ85ζ54-ζ85ζ5 | ζ8ζ54-ζ8ζ5 | -ζ8ζ53+ζ8ζ52 | ζ87ζ54-ζ87ζ5 | ζ83ζ54-ζ83ζ5 | -ζ83ζ53+ζ83ζ52 | ζ83ζ53-ζ83ζ52 | complex faithful |
| ρ23 | 2 | -2 | 0 | -2i | 2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ86ζ54+ζ86ζ5 | ζ82ζ53+ζ82ζ52 | ζ86ζ53+ζ86ζ52 | ζ82ζ54+ζ82ζ5 | ζ83ζ53-ζ83ζ52 | ζ87ζ54-ζ87ζ5 | ζ83ζ54-ζ83ζ5 | -ζ83ζ53+ζ83ζ52 | ζ85ζ54-ζ85ζ5 | ζ8ζ54-ζ8ζ5 | -ζ8ζ53+ζ8ζ52 | ζ8ζ53-ζ8ζ52 | complex faithful |
| ρ24 | 2 | -2 | 0 | 2i | -2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ82ζ53+ζ82ζ52 | ζ86ζ54+ζ86ζ5 | ζ82ζ54+ζ82ζ5 | ζ86ζ53+ζ86ζ52 | ζ85ζ54-ζ85ζ5 | -ζ8ζ53+ζ8ζ52 | ζ8ζ53-ζ8ζ52 | ζ8ζ54-ζ8ζ5 | -ζ83ζ53+ζ83ζ52 | ζ83ζ53-ζ83ζ52 | ζ83ζ54-ζ83ζ5 | ζ87ζ54-ζ87ζ5 | complex faithful |
| ρ25 | 2 | -2 | 0 | 2i | -2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ82ζ54+ζ82ζ5 | ζ86ζ53+ζ86ζ52 | ζ82ζ53+ζ82ζ52 | ζ86ζ54+ζ86ζ5 | -ζ8ζ53+ζ8ζ52 | ζ8ζ54-ζ8ζ5 | ζ85ζ54-ζ85ζ5 | ζ8ζ53-ζ8ζ52 | ζ83ζ54-ζ83ζ5 | ζ87ζ54-ζ87ζ5 | ζ83ζ53-ζ83ζ52 | -ζ83ζ53+ζ83ζ52 | complex faithful |
| ρ26 | 2 | -2 | 0 | 2i | -2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ82ζ53+ζ82ζ52 | ζ86ζ54+ζ86ζ5 | ζ82ζ54+ζ82ζ5 | ζ86ζ53+ζ86ζ52 | ζ8ζ54-ζ8ζ5 | ζ8ζ53-ζ8ζ52 | -ζ8ζ53+ζ8ζ52 | ζ85ζ54-ζ85ζ5 | ζ83ζ53-ζ83ζ52 | -ζ83ζ53+ζ83ζ52 | ζ87ζ54-ζ87ζ5 | ζ83ζ54-ζ83ζ5 | complex faithful |
►On 40 points
Generators in S40
(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)
(1 35 32 24 11)(2 36 25 17 12)(3 37 26 18 13)(4 38 27 19 14)(5 39 28 20 15)(6 40 29 21 16)(7 33 30 22 9)(8 34 31 23 10)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 29)(27 31)
G:=sub<Sym(40)| (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), (1,35,32,24,11)(2,36,25,17,12)(3,37,26,18,13)(4,38,27,19,14)(5,39,28,20,15)(6,40,29,21,16)(7,33,30,22,9)(8,34,31,23,10), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,29)(27,31)>;
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), (1,35,32,24,11)(2,36,25,17,12)(3,37,26,18,13)(4,38,27,19,14)(5,39,28,20,15)(6,40,29,21,16)(7,33,30,22,9)(8,34,31,23,10), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,29)(27,31) );
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)], [(1,35,32,24,11),(2,36,25,17,12),(3,37,26,18,13),(4,38,27,19,14),(5,39,28,20,15),(6,40,29,21,16),(7,33,30,22,9),(8,34,31,23,10)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,29),(27,31)]])
C8⋊D5 is a maximal subgroup of
D20.3C4 D5×M4(2) D20.2C4 D8⋊D5 D40⋊C2 SD16⋊D5 Q16⋊D5 C20.32D6 D30.5C4 C40⋊S3 C8⋊D25 C20.30D10 C20.31D10 C40⋊D5 D10.F5 Dic5.F5
C8⋊D5 is a maximal quotient of
C20.8Q8 C40⋊8C4 D10⋊1C8 C20.32D6 D30.5C4 C40⋊S3 C8⋊D25 C20.30D10 C20.31D10 C40⋊D5 D10.F5 Dic5.F5
Matrix representation of C8⋊D5 ►in GL2(𝔽29) generated by
| 11 | 1 |
| 12 | 18 |
| 12 | 18 |
| 13 | 22 |
| 22 | 26 |
| 16 | 7 |
G:=sub<GL(2,GF(29))| [11,12,1,18],[12,13,18,22],[22,16,26,7] >;
C8⋊D5 in GAP, Magma, Sage, TeX
C_8\rtimes D_5
% in TeX
G:=Group("C8:D5"); // GroupNames label
G:=SmallGroup(80,5);
// by ID
G=gap.SmallGroup(80,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,101,26,42,1604]);
// Polycyclic
G:=Group<a,b,c|a^8=b^5=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊D5 in TeX
Character table of C8⋊D5 in TeX