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G = D5⋊C8order 80 = 24·5

The semidirect product of D5 and C8 acting via C8/C4=C2

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

Aliases: D5⋊C8, C4.3F5, C20.3C4, D10.2C4, Dic5.4C22, C4(C5⋊C8), C5⋊C83C2, C51(C2×C8), C2.1(C2×F5), C10.1(C2×C4), (C4×D5).5C2, SmallGroup(80,28)

Series: Derived Chief Lower central Upper central

C1C5 — D5⋊C8
C1C5C10Dic5C5⋊C8 — D5⋊C8
C5 — D5⋊C8
C1C4

Generators and relations for D5⋊C8
 G = < a,b,c | a5=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a2b >

5C2
5C2
5C4
5C22
5C8
5C2×C4
5C8
5C2×C8

Character table of D5⋊C8

 class 12A2B2C4A4B4C4D58A8B8C8D8E8F8G8H1020A20B
 size 11551155455555555444
ρ111111111111111111111    trivial
ρ211-1-1-1-1111-1111-1-1-111-1-1    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ411-1-1-1-11111-1-1-1111-11-1-1    linear of order 2
ρ51111-1-1-1-11ii-i-i-i-iii1-1-1    linear of order 4
ρ61111-1-1-1-11-i-iiiii-i-i1-1-1    linear of order 4
ρ711-1-111-1-11-ii-i-iii-ii111    linear of order 4
ρ811-1-111-1-11i-iii-i-ii-i111    linear of order 4
ρ91-1-11i-ii-i1ζ8ζ85ζ83ζ87ζ83ζ87ζ85ζ8-1-ii    linear of order 8
ρ101-11-1-iii-i1ζ8ζ8ζ87ζ83ζ83ζ87ζ85ζ85-1i-i    linear of order 8
ρ111-11-1-iii-i1ζ85ζ85ζ83ζ87ζ87ζ83ζ8ζ8-1i-i    linear of order 8
ρ121-1-11i-ii-i1ζ85ζ8ζ87ζ83ζ87ζ83ζ8ζ85-1-ii    linear of order 8
ρ131-11-1i-i-ii1ζ87ζ87ζ8ζ85ζ85ζ8ζ83ζ83-1-ii    linear of order 8
ρ141-11-1i-i-ii1ζ83ζ83ζ85ζ8ζ8ζ85ζ87ζ87-1-ii    linear of order 8
ρ151-1-11-ii-ii1ζ83ζ87ζ8ζ85ζ8ζ85ζ87ζ83-1i-i    linear of order 8
ρ161-1-11-ii-ii1ζ87ζ83ζ85ζ8ζ85ζ8ζ83ζ87-1i-i    linear of order 8
ρ1744004400-100000000-1-1-1    orthogonal lifted from F5
ρ184400-4-400-100000000-111    orthogonal lifted from C2×F5
ρ194-400-4i4i00-1000000001-ii    complex faithful, Schur index 2
ρ204-4004i-4i00-1000000001i-i    complex faithful, Schur index 2

Smallest permutation representation of D5⋊C8
On 40 points
Generators in S40
(1 35 9 22 26)(2 23 36 27 10)(3 28 24 11 37)(4 12 29 38 17)(5 39 13 18 30)(6 19 40 31 14)(7 32 20 15 33)(8 16 25 34 21)
(1 26)(2 10)(3 37)(4 17)(5 30)(6 14)(7 33)(8 21)(11 28)(12 38)(15 32)(16 34)(18 39)(19 31)(22 35)(23 27)
(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)| (1,35,9,22,26)(2,23,36,27,10)(3,28,24,11,37)(4,12,29,38,17)(5,39,13,18,30)(6,19,40,31,14)(7,32,20,15,33)(8,16,25,34,21), (1,26)(2,10)(3,37)(4,17)(5,30)(6,14)(7,33)(8,21)(11,28)(12,38)(15,32)(16,34)(18,39)(19,31)(22,35)(23,27), (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( (1,35,9,22,26)(2,23,36,27,10)(3,28,24,11,37)(4,12,29,38,17)(5,39,13,18,30)(6,19,40,31,14)(7,32,20,15,33)(8,16,25,34,21), (1,26)(2,10)(3,37)(4,17)(5,30)(6,14)(7,33)(8,21)(11,28)(12,38)(15,32)(16,34)(18,39)(19,31)(22,35)(23,27), (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([(1,35,9,22,26),(2,23,36,27,10),(3,28,24,11,37),(4,12,29,38,17),(5,39,13,18,30),(6,19,40,31,14),(7,32,20,15,33),(8,16,25,34,21)], [(1,26),(2,10),(3,37),(4,17),(5,30),(6,14),(7,33),(8,21),(11,28),(12,38),(15,32),(16,34),(18,39),(19,31),(22,35),(23,27)], [(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)])

D5⋊C8 is a maximal subgroup of
C8×F5  C8⋊F5  D20⋊C4  Q8⋊F5  D5⋊M4(2)  D4.F5  Q8.F5  D15⋊C8  C60.C4  D25⋊C8  Dic5.4F5  D10.2F5  C20.14F5  C20.F5  C20.11F5  A5⋊C8  C4.6S5
D5⋊C8 is a maximal quotient of
D5⋊C16  C8.F5  C4×C5⋊C8  D10⋊C8  Dic5⋊C8  D15⋊C8  C60.C4  D25⋊C8  Dic5.4F5  D10.2F5  C20.14F5  C20.F5  C20.11F5

Matrix representation of D5⋊C8 in GL4(𝔽41) generated by

0100
0010
0001
40404040
,
0100
1000
40404040
0001
,
3201515
1515032
2617260
924249
G:=sub<GL(4,GF(41))| [0,0,0,40,1,0,0,40,0,1,0,40,0,0,1,40],[0,1,40,0,1,0,40,0,0,0,40,0,0,0,40,1],[32,15,26,9,0,15,17,24,15,0,26,24,15,32,0,9] >;

D5⋊C8 in GAP, Magma, Sage, TeX

D_5\rtimes C_8
% in TeX

G:=Group("D5:C8");
// GroupNames label

G:=SmallGroup(80,28);
// by ID

G=gap.SmallGroup(80,28);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,46,42,804,414]);
// Polycyclic

G:=Group<a,b,c|a^5=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b>;
// generators/relations

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Subgroup lattice of D5⋊C8 in TeX
Character table of D5⋊C8 in TeX

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