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## G = Q16×F5order 320 = 26·5

### Direct product of Q16 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q16×F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — Q8×F5 — Q16×F5
 Lower central C5 — C10 — C20 — Q16×F5
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16×F5
G = < a,b,c,d | a8=c5=d4=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 418 in 110 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×2], C4, C4 [×10], C22, C5, C8, C8 [×2], C2×C4 [×7], Q8 [×2], Q8 [×4], D5 [×2], C10, C42 [×3], C4⋊C4 [×4], C2×C8 [×2], Q16, Q16 [×3], C2×Q8 [×2], Dic5, Dic5 [×2], C20, C20 [×2], F5 [×2], F5 [×3], D10, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C52C8, C40, C5⋊C8, Dic10 [×2], Dic10 [×2], C4×D5, C4×D5 [×2], C5×Q8 [×2], C2×F5 [×2], C2×F5 [×2], C4×Q16, C8×D5, Dic20, C5⋊Q16 [×2], C5×Q16, D5⋊C8, C4×F5, C4×F5 [×2], C4⋊F5 [×2], C4⋊F5 [×2], Q8×D5 [×2], C8×F5, D5.D8, Q8⋊F5 [×2], D5×Q16, Q8×F5 [×2], Q16×F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, Q16 [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×Q16, C4○D8, C2×F5 [×3], C4×Q16, C22×F5, D4×F5, Q16×F5

Smallest permutation representation of Q16×F5
On 80 points
Generators in S80
(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)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 75 41 12 66)(2 76 42 13 67)(3 77 43 14 68)(4 78 44 15 69)(5 79 45 16 70)(6 80 46 9 71)(7 73 47 10 72)(8 74 48 11 65)(17 40 50 60 29)(18 33 51 61 30)(19 34 52 62 31)(20 35 53 63 32)(21 36 54 64 25)(22 37 55 57 26)(23 38 56 58 27)(24 39 49 59 28)
(1 60 5 64)(2 61 6 57)(3 62 7 58)(4 63 8 59)(9 26 42 51)(10 27 43 52)(11 28 44 53)(12 29 45 54)(13 30 46 55)(14 31 47 56)(15 32 48 49)(16 25 41 50)(17 70 36 75)(18 71 37 76)(19 72 38 77)(20 65 39 78)(21 66 40 79)(22 67 33 80)(23 68 34 73)(24 69 35 74)

G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,45,21,41)(18,44,22,48)(19,43,23,47)(20,42,24,46)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,75,41,12,66)(2,76,42,13,67)(3,77,43,14,68)(4,78,44,15,69)(5,79,45,16,70)(6,80,46,9,71)(7,73,47,10,72)(8,74,48,11,65)(17,40,50,60,29)(18,33,51,61,30)(19,34,52,62,31)(20,35,53,63,32)(21,36,54,64,25)(22,37,55,57,26)(23,38,56,58,27)(24,39,49,59,28), (1,60,5,64)(2,61,6,57)(3,62,7,58)(4,63,8,59)(9,26,42,51)(10,27,43,52)(11,28,44,53)(12,29,45,54)(13,30,46,55)(14,31,47,56)(15,32,48,49)(16,25,41,50)(17,70,36,75)(18,71,37,76)(19,72,38,77)(20,65,39,78)(21,66,40,79)(22,67,33,80)(23,68,34,73)(24,69,35,74)>;

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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,45,21,41)(18,44,22,48)(19,43,23,47)(20,42,24,46)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,75,41,12,66)(2,76,42,13,67)(3,77,43,14,68)(4,78,44,15,69)(5,79,45,16,70)(6,80,46,9,71)(7,73,47,10,72)(8,74,48,11,65)(17,40,50,60,29)(18,33,51,61,30)(19,34,52,62,31)(20,35,53,63,32)(21,36,54,64,25)(22,37,55,57,26)(23,38,56,58,27)(24,39,49,59,28), (1,60,5,64)(2,61,6,57)(3,62,7,58)(4,63,8,59)(9,26,42,51)(10,27,43,52)(11,28,44,53)(12,29,45,54)(13,30,46,55)(14,31,47,56)(15,32,48,49)(16,25,41,50)(17,70,36,75)(18,71,37,76)(19,72,38,77)(20,65,39,78)(21,66,40,79)(22,67,33,80)(23,68,34,73)(24,69,35,74) );

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),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,60,5,64),(2,59,6,63),(3,58,7,62),(4,57,8,61),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(17,45,21,41),(18,44,22,48),(19,43,23,47),(20,42,24,46),(25,75,29,79),(26,74,30,78),(27,73,31,77),(28,80,32,76),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)], [(1,75,41,12,66),(2,76,42,13,67),(3,77,43,14,68),(4,78,44,15,69),(5,79,45,16,70),(6,80,46,9,71),(7,73,47,10,72),(8,74,48,11,65),(17,40,50,60,29),(18,33,51,61,30),(19,34,52,62,31),(20,35,53,63,32),(21,36,54,64,25),(22,37,55,57,26),(23,38,56,58,27),(24,39,49,59,28)], [(1,60,5,64),(2,61,6,57),(3,62,7,58),(4,63,8,59),(9,26,42,51),(10,27,43,52),(11,28,44,53),(12,29,45,54),(13,30,46,55),(14,31,47,56),(15,32,48,49),(16,25,41,50),(17,70,36,75),(18,71,37,76),(19,72,38,77),(20,65,39,78),(21,66,40,79),(22,67,33,80),(23,68,34,73),(24,69,35,74)])

35 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4P 5 8A 8B 8C ··· 8H 10 20A 20B 20C 40A 40B order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 ··· 4 5 8 8 8 ··· 8 10 20 20 20 40 40 size 1 1 5 5 2 4 4 5 5 5 5 10 10 10 20 ··· 20 4 2 2 10 ··· 10 4 8 16 16 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 Q16 C4○D8 F5 C2×F5 C2×F5 D4×F5 Q16×F5 kernel Q16×F5 C8×F5 D5.D8 Q8⋊F5 D5×Q16 Q8×F5 Dic20 C5⋊Q16 C5×Q16 C2×F5 Dic5 F5 D5 Q16 C8 Q8 C2 C1 # reps 1 1 1 2 1 2 2 4 2 2 2 4 4 1 1 2 1 2

Matrix representation of Q16×F5 in GL6(𝔽41)

 0 17 0 0 0 0 12 17 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
,
 9 0 0 0 0 0 9 32 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
,
 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
,
 40 0 0 0 0 0 0 40 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))| [0,12,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

Q16×F5 in GAP, Magma, Sage, TeX

Q_{16}\times F_5
% in TeX

G:=Group("Q16xF5");
// GroupNames label

G:=SmallGroup(320,1076);
// by ID

G=gap.SmallGroup(320,1076);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,851,438,102,6278,1595]);
// Polycyclic

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