Q8xF5 - GroupNames

G = Q₈×F₅ order 160 = 2⁵·5

Direct product of Q₈ and F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈×F₅, Dic₁₀⋊₃C₄, D₁₀.₁₃C₂³, C₅⋊(C₄×Q₈), (C₅×Q₈)⋊₃C₄, C₄⋊F₅.₂C₂, C₄.₇(C₂×F₅), C₂₀.₇(C₂×C₄), (C₄×F₅).₁C₂, D₅.₂(C₂×Q₈), (Q₈×D₅).₃C₂, D₅.₃(C₄○D₄), Dic₅.₂(C₂×C₄), (C₂×F₅).₄C₂², C₂.₁₁(C₂²×F₅), C₁₀.₁₀(C₂²×C₄), (C₄×D₅).₁₄C₂², SmallGroup(160,209)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — Q₈×F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂×F₅ — C₄×F₅ — Q₈×F₅

Lower central

C₅ — C₁₀ — Q₈×F₅

Upper central

C₁ — C₂ — Q₈

Generators and relations for Q₈×F₅
G = < a,b,c,d | a⁴=c⁵=d⁴=1, b²=a², bab^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Subgroups: 204 in 70 conjugacy classes, 38 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₂×C₄, Q₈, Q₈, D₅, C₁₀, C₄², C₄⋊C₄, C₂×Q₈, Dic₅, C₂₀, F₅, F₅, D₁₀, C₄×Q₈, Dic₁₀, C₄×D₅, C₅×Q₈, C₂×F₅, C₂×F₅, C₄×F₅, C₄⋊F₅, Q₈×D₅, Q₈×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, Q₈, C₂³, C₂²×C₄, C₂×Q₈, C₄○D₄, F₅, C₄×Q₈, C₂×F₅, C₂²×F₅, Q₈×F₅

Character table of Q₈×F₅

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	5	10	20A	20B	20C
size	1	1	5	5	2	2	2	5	5	5	5	10	10	10	10	10	10	10	10	10	4	4	8	8	8
ρ₁	1	1	1	1	1	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	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	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	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	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	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	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	1	1	-1	1	-1	linear of order 2
ρ₉	1	1	-1	-1	1	1	1	-i	i	i	-i	-i	-1	-1	-1	i	-i	i	-i	i	1	1	1	1	1	linear of order 4
ρ₁₀	1	1	-1	-1	-1	-1	1	-i	i	i	-i	i	1	1	-1	i	-i	-i	i	-i	1	1	-1	-1	1	linear of order 4
ρ₁₁	1	1	-1	-1	-1	1	-1	i	-i	-i	i	-i	-1	1	1	i	-i	-i	i	i	1	1	1	-1	-1	linear of order 4
ρ₁₂	1	1	-1	-1	1	-1	-1	i	-i	-i	i	i	1	-1	1	i	-i	i	-i	-i	1	1	-1	1	-1	linear of order 4
ρ₁₃	1	1	-1	-1	-1	1	-1	-i	i	i	-i	i	-1	1	1	-i	i	i	-i	-i	1	1	1	-1	-1	linear of order 4
ρ₁₄	1	1	-1	-1	1	-1	-1	-i	i	i	-i	-i	1	-1	1	-i	i	-i	i	i	1	1	-1	1	-1	linear of order 4
ρ₁₅	1	1	-1	-1	1	1	1	i	-i	-i	i	i	-1	-1	-1	-i	i	-i	i	-i	1	1	1	1	1	linear of order 4
ρ₁₆	1	1	-1	-1	-1	-1	1	i	-i	-i	i	-i	1	1	-1	-i	i	i	-i	i	1	1	-1	-1	1	linear of order 4
ρ₁₇	2	-2	2	-2	0	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	2	-2	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₈	2	-2	2	-2	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	2	-2	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₉	2	-2	-2	2	0	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	2	-2	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	0	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	2	-2	0	0	0	complex lifted from C₄○D₄
ρ₂₁	4	4	0	0	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	-1	1	-1	1	orthogonal lifted from C₂×F₅
ρ₂₂	4	4	0	0	4	4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₂₃	4	4	0	0	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	-1	-1	1	1	orthogonal lifted from C₂×F₅
ρ₂₄	4	4	0	0	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	-1	1	1	-1	orthogonal lifted from C₂×F₅
ρ₂₅	8	-8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	2	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of Q₈×F₅
►On 40 points

Generators in S₄₀

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

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

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

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

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

Q₈×F₅ is a maximal subgroup of
SD₁₆⋊F₅ Dic₂₀⋊C₄ D₅.2_-¹⁺⁴ D₅.2₊¹⁺⁴ Dic₆⋊₅F₅
Q₈×F₅ is a maximal quotient of
Dic₁₀⋊C₈ Dic₅.M₄(2) C₂₀.M₄(2) C₄⋊C₄⋊₅F₅ C₂₀⋊(C₄⋊C₄) C₂₀.₆M₄(2) (C₂×F₅)⋊Q₈ Dic₆⋊₅F₅

Matrix representation of Q₈×F₅ ►in GL₆(𝔽₄₁)

0	1	0	0	0	0
40	0	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

9	0	0	0	0	0
0	32	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	40
0	0	1	0	0	40
0	0	0	1	0	40
0	0	0	0	1	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	40	0
0	0	40	0	0	0
0	0	0	0	0	40
0	0	0	40	0	0

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,0,40,0] >;

Q₈×F₅ in GAP, Magma, Sage, TeX

Q_8\times F_5

% in TeX

G:=Group("Q8xF5");

// GroupNames label

G:=SmallGroup(160,209);

// by ID

G=gap.SmallGroup(160,209);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,2309,599]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^5=d^4=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^-1=c^3>;

// generators/relations

Export

Character table of Q₈×F₅ in TeX

G = Q8×F5 order 160 = 25·5

Direct product of Q8 and F5

G = Q₈×F₅ order 160 = 2⁵·5

Direct product of Q₈ and F₅