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## G = Q8×F5order 160 = 25·5

### Direct product of Q8 and F5

Aliases: Q8×F5, Dic103C4, 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 [×2], C4 [×3], C4 [×8], C22, C5, C2×C4 [×7], Q8, Q8 [×3], D5 [×2], C10, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×3], C20 [×3], F5 [×2], F5 [×3], D10, C4×Q8, Dic10 [×3], C4×D5 [×3], C5×Q8, C2×F5, C2×F5 [×3], C4×F5 [×3], C4⋊F5 [×3], Q8×D5, Q8×F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C2×F5 [×3], 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  Dic65F5
Q8×F5 is a maximal quotient of
Dic10⋊C8  Dic5.M4(2)  C20.M4(2)  C4⋊C45F5  C20⋊(C4⋊C4)  C20.6M4(2)  (C2×F5)⋊Q8  Dic65F5

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

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