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