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G = C5⋊C8order 40 = 23·5

The semidirect product of C5 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5⋊C8, C2.F5, C10.C4, Dic5.2C2, SmallGroup(40,3)

Series: Derived Chief Lower central Upper central

C1C5 — C5⋊C8
C1C5C10Dic5 — C5⋊C8
C5 — C5⋊C8
C1C2

Generators and relations for C5⋊C8
 G = < a,b | a5=b8=1, bab-1=a3 >

5C4
5C8

Character table of C5⋊C8

 class 124A4B58A8B8C8D10
 size 1155455554
ρ11111111111    trivial
ρ211111-1-1-1-11    linear of order 2
ρ311-1-11i-ii-i1    linear of order 4
ρ411-1-11-ii-ii1    linear of order 4
ρ51-1i-i1ζ85ζ87ζ8ζ83-1    linear of order 8
ρ61-1-ii1ζ83ζ8ζ87ζ85-1    linear of order 8
ρ71-1i-i1ζ8ζ83ζ85ζ87-1    linear of order 8
ρ81-1-ii1ζ87ζ85ζ83ζ8-1    linear of order 8
ρ94400-10000-1    orthogonal lifted from F5
ρ104-400-100001    symplectic faithful, Schur index 2

Smallest permutation representation of C5⋊C8
Regular action on 40 points
Generators in S40
(1 35 22 27 14)(2 28 36 15 23)(3 16 29 24 37)(4 17 9 38 30)(5 39 18 31 10)(6 32 40 11 19)(7 12 25 20 33)(8 21 13 34 26)
(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)

G:=sub<Sym(40)| (1,35,22,27,14)(2,28,36,15,23)(3,16,29,24,37)(4,17,9,38,30)(5,39,18,31,10)(6,32,40,11,19)(7,12,25,20,33)(8,21,13,34,26), (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)>;

G:=Group( (1,35,22,27,14)(2,28,36,15,23)(3,16,29,24,37)(4,17,9,38,30)(5,39,18,31,10)(6,32,40,11,19)(7,12,25,20,33)(8,21,13,34,26), (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) );

G=PermutationGroup([(1,35,22,27,14),(2,28,36,15,23),(3,16,29,24,37),(4,17,9,38,30),(5,39,18,31,10),(6,32,40,11,19),(7,12,25,20,33),(8,21,13,34,26)], [(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)])

Matrix representation of C5⋊C8 in GL4(𝔽3) generated by

0212
1111
1220
0202
,
2020
0102
2000
2200
G:=sub<GL(4,GF(3))| [0,1,1,0,2,1,2,2,1,1,2,0,2,1,0,2],[2,0,2,2,0,1,0,2,2,0,0,0,0,2,0,0] >;

C5⋊C8 in GAP, Magma, Sage, TeX

C_5\rtimes C_8
% in TeX

G:=Group("C5:C8");
// GroupNames label

G:=SmallGroup(40,3);
// by ID

G=gap.SmallGroup(40,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,8,21,259,263]);
// Polycyclic

G:=Group<a,b|a^5=b^8=1,b*a*b^-1=a^3>;
// generators/relations

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