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G = C52C8order 40 = 23·5

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

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

Aliases: C52C8, C4.2D5, C2.Dic5, C10.2C4, C20.2C2, SmallGroup(40,1)

Series: Derived Chief Lower central Upper central

C1C5 — C52C8
C1C5C10C20 — C52C8
C5 — C52C8
C1C4

Generators and relations for C52C8
 G = < a,b | a5=b8=1, bab-1=a-1 >

5C8

Character table of C52C8

 class 124A4B5A5B8A8B8C8D10A10B20A20B20C20D
 size 1111225555222222
ρ11111111111111111    trivial
ρ2111111-1-1-1-1111111    linear of order 2
ρ311-1-111-ii-ii11-1-1-1-1    linear of order 4
ρ411-1-111i-ii-i11-1-1-1-1    linear of order 4
ρ51-1i-i11ζ87ζ85ζ83ζ8-1-1i-ii-i    linear of order 8
ρ61-1i-i11ζ83ζ8ζ87ζ85-1-1i-ii-i    linear of order 8
ρ71-1-ii11ζ8ζ83ζ85ζ87-1-1-ii-ii    linear of order 8
ρ81-1-ii11ζ85ζ87ζ8ζ83-1-1-ii-ii    linear of order 8
ρ92222-1-5/2-1+5/20000-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ102222-1+5/2-1-5/20000-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ1122-2-2-1+5/2-1-5/20000-1+5/2-1-5/21+5/21+5/21-5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ1222-2-2-1-5/2-1+5/20000-1-5/2-1+5/21-5/21-5/21+5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ132-2-2i2i-1+5/2-1-5/200001-5/21+5/2ζ43ζ5343ζ52ζ4ζ534ζ52ζ43ζ5443ζ5ζ4ζ544ζ5    complex faithful, Schur index 2
ρ142-22i-2i-1-5/2-1+5/200001+5/21-5/2ζ4ζ544ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ43ζ5343ζ52    complex faithful, Schur index 2
ρ152-2-2i2i-1-5/2-1+5/200001+5/21-5/2ζ43ζ5443ζ5ζ4ζ544ζ5ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful, Schur index 2
ρ162-22i-2i-1+5/2-1-5/200001-5/21+5/2ζ4ζ534ζ52ζ43ζ5343ζ52ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful, Schur index 2

Smallest permutation representation of C52C8
Regular action on 40 points
Generators in S40
(1 35 27 20 11)(2 12 21 28 36)(3 37 29 22 13)(4 14 23 30 38)(5 39 31 24 15)(6 16 17 32 40)(7 33 25 18 9)(8 10 19 26 34)
(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)| (1,35,27,20,11)(2,12,21,28,36)(3,37,29,22,13)(4,14,23,30,38)(5,39,31,24,15)(6,16,17,32,40)(7,33,25,18,9)(8,10,19,26,34), (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( (1,35,27,20,11)(2,12,21,28,36)(3,37,29,22,13)(4,14,23,30,38)(5,39,31,24,15)(6,16,17,32,40)(7,33,25,18,9)(8,10,19,26,34), (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([(1,35,27,20,11),(2,12,21,28,36),(3,37,29,22,13),(4,14,23,30,38),(5,39,31,24,15),(6,16,17,32,40),(7,33,25,18,9),(8,10,19,26,34)], [(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 C52C8 in GL2(𝔽29) generated by

1121
912
,
017
10
G:=sub<GL(2,GF(29))| [11,9,21,12],[0,1,17,0] >;

C52C8 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_8
% in TeX

G:=Group("C5:2C8");
// GroupNames label

G:=SmallGroup(40,1);
// by ID

G=gap.SmallGroup(40,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,8,21,515]);
// Polycyclic

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

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