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