metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3F5, C40⋊4C4, D5.M4(2), C10.2C42, C5⋊C8⋊1C4, C5⋊2C8⋊8C4, (C2×F5).C4, C5⋊1(C8⋊C4), C2.3(C4×F5), D5⋊C8.2C2, (C4×F5).2C2, C4.17(C2×F5), C20.16(C2×C4), (C8×D5).10C2, D10.6(C2×C4), Dic5.8(C2×C4), (C4×D5).32C22, SmallGroup(160,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊F5
G = < a,b,c | a8=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >
Character table of C8⋊F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 2 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 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 | 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 | -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 | -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 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | -i | i | -1 | -i | -1 | 1 | 1 | i | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | i | -i | i | -i | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | i | -i | -1 | i | -1 | 1 | 1 | -i | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | i | -i | 1 | i | 1 | -1 | -1 | -i | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | -i | i | 1 | -i | 1 | -1 | -1 | i | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | -i | -i | i | -i | i | i | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | -i | 1 | 1 | 1 | -i | -1 | i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | i | 1 | -1 | -1 | -i | 1 | i | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | i | i | -i | i | -i | -i | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -i | i | -i | i | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | -i | 1 | -1 | -1 | i | 1 | -i | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | i | 1 | 1 | 1 | i | -1 | -i | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ17 | 2 | -2 | -2 | 2 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | 2 | -2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | -2 | 2 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | 2 | -2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ23 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | i | -i | -i | i | complex lifted from C4×F5 |
| ρ24 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -i | i | i | -i | complex lifted from C4×F5 |
| ρ25 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | i | -i | 2ζ8ζ54+2ζ8ζ5+ζ8 | 2ζ83ζ53+2ζ83ζ52+ζ83 | 2ζ83ζ54+2ζ83ζ5+ζ83 | 2ζ85ζ54+2ζ85ζ5+ζ85 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -i | i | 2ζ83ζ53+2ζ83ζ52+ζ83 | 2ζ8ζ54+2ζ8ζ5+ζ8 | 2ζ85ζ54+2ζ85ζ5+ζ85 | 2ζ83ζ54+2ζ83ζ5+ζ83 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -i | i | 2ζ83ζ54+2ζ83ζ5+ζ83 | 2ζ85ζ54+2ζ85ζ5+ζ85 | 2ζ8ζ54+2ζ8ζ5+ζ8 | 2ζ83ζ53+2ζ83ζ52+ζ83 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | i | -i | 2ζ85ζ54+2ζ85ζ5+ζ85 | 2ζ83ζ54+2ζ83ζ5+ζ83 | 2ζ83ζ53+2ζ83ζ52+ζ83 | 2ζ8ζ54+2ζ8ζ5+ζ8 | 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 11 24 32 35)(2 12 17 25 36)(3 13 18 26 37)(4 14 19 27 38)(5 15 20 28 39)(6 16 21 29 40)(7 9 22 30 33)(8 10 23 31 34)
(1 7 5 3)(2 4 6 8)(9 20 37 32)(10 17 38 29)(11 22 39 26)(12 19 40 31)(13 24 33 28)(14 21 34 25)(15 18 35 30)(16 23 36 27)
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), (1,11,24,32,35)(2,12,17,25,36)(3,13,18,26,37)(4,14,19,27,38)(5,15,20,28,39)(6,16,21,29,40)(7,9,22,30,33)(8,10,23,31,34), (1,7,5,3)(2,4,6,8)(9,20,37,32)(10,17,38,29)(11,22,39,26)(12,19,40,31)(13,24,33,28)(14,21,34,25)(15,18,35,30)(16,23,36,27)>;
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), (1,11,24,32,35)(2,12,17,25,36)(3,13,18,26,37)(4,14,19,27,38)(5,15,20,28,39)(6,16,21,29,40)(7,9,22,30,33)(8,10,23,31,34), (1,7,5,3)(2,4,6,8)(9,20,37,32)(10,17,38,29)(11,22,39,26)(12,19,40,31)(13,24,33,28)(14,21,34,25)(15,18,35,30)(16,23,36,27) );
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)], [(1,11,24,32,35),(2,12,17,25,36),(3,13,18,26,37),(4,14,19,27,38),(5,15,20,28,39),(6,16,21,29,40),(7,9,22,30,33),(8,10,23,31,34)], [(1,7,5,3),(2,4,6,8),(9,20,37,32),(10,17,38,29),(11,22,39,26),(12,19,40,31),(13,24,33,28),(14,21,34,25),(15,18,35,30),(16,23,36,27)]])
C8⋊F5 is a maximal subgroup of
C16⋊F5 C16⋊4F5 C20.12C42 M4(2)×F5 M4(2)⋊5F5 D40⋊C4 D8⋊F5 SD16⋊F5 SD16⋊2F5 Dic20⋊C4 Q16⋊F5 C30.3C42 C30.4C42 C24⋊F5
C8⋊F5 is a maximal quotient of
C16⋊F5 C16⋊4F5 C40⋊C8 C20.31M4(2) D10.3M4(2) C30.3C42 C30.4C42 C24⋊F5
Matrix representation of C8⋊F5 ►in GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
C8⋊F5 in GAP, Magma, Sage, TeX
C_8\rtimes F_5
% in TeX
G:=Group("C8:F5"); // GroupNames label
G:=SmallGroup(160,67);
// by ID
G=gap.SmallGroup(160,67);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c|a^8=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C8⋊F5 in TeX
Character table of C8⋊F5 in TeX