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## G = D8×F5order 320 = 26·5

### Direct product of D8 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8×F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — D4×F5 — D8×F5
 Lower central C5 — C10 — C20 — D8×F5
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8×F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 674 in 134 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×6], C4, C4 [×6], C22 [×9], C5, C8, C8 [×2], C2×C4 [×9], D4 [×2], D4 [×4], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8, D8 [×3], C22×C4 [×2], C2×D4 [×2], Dic5, C20, F5 [×2], F5 [×3], D10, D10 [×6], C2×C10 [×2], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C52C8, C40, C5⋊C8, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5 [×2], C2×F5 [×6], C22×D5 [×2], C4×D8, C8×D5, D40, D4⋊D5 [×2], C5×D8, D5⋊C8, C4×F5, C4⋊F5 [×2], C22⋊F5 [×2], D4×D5 [×2], C22×F5 [×2], C8×F5, D5.D8, D20⋊C4 [×2], D5×D8, D4×F5 [×2], D8×F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×D8, C4○D8, C2×F5 [×3], C4×D8, C22×F5, D4×F5, D8×F5

Smallest permutation representation of D8×F5
On 40 points
Generators in S40
(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 5)(9 12)(10 11)(13 16)(14 15)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)
(1 11 25 22 35)(2 12 26 23 36)(3 13 27 24 37)(4 14 28 17 38)(5 15 29 18 39)(6 16 30 19 40)(7 9 31 20 33)(8 10 32 21 34)
(9 31 33 20)(10 32 34 21)(11 25 35 22)(12 26 36 23)(13 27 37 24)(14 28 38 17)(15 29 39 18)(16 30 40 19)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 8A 8B 8C ··· 8H 10A 10B 10C 20 40A 40B order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 8 8 8 ··· 8 10 10 10 20 40 40 size 1 1 4 4 5 5 20 20 2 5 5 5 5 10 10 10 20 20 20 20 4 2 2 10 ··· 10 4 16 16 8 8 8

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 D8 C4○D8 F5 C2×F5 C2×F5 D4×F5 D8×F5 kernel D8×F5 C8×F5 D5.D8 D20⋊C4 D5×D8 D4×F5 D40 D4⋊D5 C5×D8 C2×F5 Dic5 F5 D5 D8 C8 D4 C2 C1 # reps 1 1 1 2 1 2 2 4 2 2 2 4 4 1 1 2 1 2

Matrix representation of D8×F5 in GL6(𝔽41)

 0 24 0 0 0 0 29 24 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 24 0 0 0 0 12 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
,
 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
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [0,29,0,0,0,0,24,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,24,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D8×F5 in GAP, Magma, Sage, TeX

D_8\times F_5
% in TeX

G:=Group("D8xF5");
// GroupNames label

G:=SmallGroup(320,1068);
// by ID

G=gap.SmallGroup(320,1068);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,851,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=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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