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## G = Q8×D5order 80 = 24·5

### Direct product of Q8 and D5

Aliases: Q8×D5, C4.6D10, Dic104C2, C20.6C22, C10.7C23, D10.9C22, Dic5.3C22, C52(C2×Q8), (C5×Q8)⋊2C2, (C4×D5).1C2, C2.8(C22×D5), SmallGroup(80,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8×D5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — Q8×D5
 Lower central C5 — C10 — Q8×D5
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D5
G = < a,b,c,d | a4=c5=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of Q8×D5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 10A 10B 20A 20B 20C 20D 20E 20F size 1 1 5 5 2 2 2 10 10 10 2 2 2 2 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ6 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 linear of order 2 ρ7 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ9 2 2 0 0 2 2 2 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 2 0 0 -2 -2 2 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ11 2 2 0 0 2 -2 -2 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ12 2 2 0 0 2 -2 -2 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ13 2 2 0 0 2 2 2 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ14 2 2 0 0 -2 2 -2 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 orthogonal lifted from D10 ρ15 2 2 0 0 -2 2 -2 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 orthogonal lifted from D10 ρ16 2 2 0 0 -2 -2 2 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ17 2 -2 2 -2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 -2 2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 1+√5 1-√5 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ20 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 1-√5 1+√5 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8×D5
On 40 points
Generators in S40
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(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)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)

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

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

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

Q8×D5 is a maximal subgroup of
Q8⋊F5  SD16⋊D5  Q16⋊D5  Q8.10D10  D4.10D10  D15⋊Q8  Dic10⋊D5
Q8×D5 is a maximal quotient of
Dic53Q8  C20⋊Q8  Dic5.Q8  D10⋊Q8  D102Q8  Dic5⋊Q8  D103Q8  D15⋊Q8  Dic10⋊D5

Matrix representation of Q8×D5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 5 0 0 16 40
,
 40 0 0 0 0 40 0 0 0 0 21 29 0 0 30 20
,
 7 1 0 0 33 40 0 0 0 0 1 0 0 0 0 1
,
 40 40 0 0 0 1 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,5,40],[40,0,0,0,0,40,0,0,0,0,21,30,0,0,29,20],[7,33,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,40,1,0,0,0,0,40,0,0,0,0,40] >;

Q8×D5 in GAP, Magma, Sage, TeX

Q_8\times D_5
% in TeX

G:=Group("Q8xD5");
// GroupNames label

G:=SmallGroup(80,41);
// by ID

G=gap.SmallGroup(80,41);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,46,97,42,1604]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^5=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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