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G = C5⋊2C8  order 40 = 23·5

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

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

Aliases: C5⋊2C8, C4.2D5, C2.Dic5, C10.2C4, C20.2C2, SmallGroup(40,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C5⋊2C8
Chief series
C1 — C5 — C10 — C20 — C5⋊2C8
Lower central
C5 — C5⋊2C8
Upper central
C1 — C4

Generators and relations for C5⋊2C8
 G = < a,b | a5=b8=1, bab-1=a-1 >

C1
C2
C5
C4
C10
5C8
C20
C5⋊2C8

Character table of C5⋊2C8

 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+√5/2ζ43ζ53+ζ43ζ52ζ4ζ53+ζ4ζ52ζ43ζ54+ζ43ζ5ζ4ζ54+ζ4ζ5    complex faithful, Schur index 2
ρ142-22i-2i-1-√5/2-1+√5/200001+√5/21-√5/2ζ4ζ54+ζ4ζ5ζ43ζ54+ζ43ζ5ζ4ζ53+ζ4ζ52ζ43ζ53+ζ43ζ52    complex faithful, Schur index 2
ρ152-2-2i2i-1-√5/2-1+√5/200001+√5/21-√5/2ζ43ζ54+ζ43ζ5ζ4ζ54+ζ4ζ5ζ43ζ53+ζ43ζ52ζ4ζ53+ζ4ζ52    complex faithful, Schur index 2
ρ162-22i-2i-1+√5/2-1-√5/200001-√5/21+√5/2ζ4ζ53+ζ4ζ52ζ43ζ53+ζ43ζ52ζ4ζ54+ζ4ζ5ζ43ζ54+ζ43ζ5    complex faithful, Schur index 2

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

C5⋊2C8 is a maximal subgroup of
 C5⋊C16  C8×D5  C8⋊D5  C4.Dic5  D4⋊D5  D4.D5  Q8⋊D5  C5⋊Q16  C15⋊3C8  C25⋊2C8  C52⋊7C8  C52⋊3C8  C35⋊3C8  (C3×C15)⋊9C8  C5⋊2F9  C55⋊3C8
C5⋊2C8 is a maximal quotient of
 C5⋊2C16  C15⋊3C8  C25⋊2C8  C52⋊7C8  C52⋊3C8  C35⋊3C8  (C3×C15)⋊9C8  C5⋊2F9  C55⋊3C8

Matrix representation of C5⋊2C8 ►in GL2(𝔽29) generated by

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

C5⋊2C8 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
 

Export

Subgroup lattice of C5⋊2C8 in TeX
Character table of C5⋊2C8 in TeX

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