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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, C4, C4, C22, C5, C8, C8, C2×C4, Q8, Q8, D5, C10, C42, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, F5, F5, D10, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C52C8, C40, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×F5, C4×Q16, C8×D5, Dic20, C5⋊Q16, C5×Q16, D5⋊C8, C4×F5, C4×F5, C4⋊F5, C4⋊F5, Q8×D5, C8×F5, D5.D8, Q8⋊F5, D5×Q16, Q8×F5, Q16×F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, Q16, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×Q16, C4○D8, C2×F5, 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 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 38 21 34)(18 37 22 33)(19 36 23 40)(20 35 24 39)(41 74 45 78)(42 73 46 77)(43 80 47 76)(44 79 48 75)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 10 34 77 66)(2 11 35 78 67)(3 12 36 79 68)(4 13 37 80 69)(5 14 38 73 70)(6 15 39 74 71)(7 16 40 75 72)(8 9 33 76 65)(17 42 50 60 29)(18 43 51 61 30)(19 44 52 62 31)(20 45 53 63 32)(21 46 54 64 25)(22 47 55 57 26)(23 48 56 58 27)(24 41 49 59 28)
(1 60 5 64)(2 61 6 57)(3 62 7 58)(4 63 8 59)(9 24 69 45)(10 17 70 46)(11 18 71 47)(12 19 72 48)(13 20 65 41)(14 21 66 42)(15 22 67 43)(16 23 68 44)(25 34 50 73)(26 35 51 74)(27 36 52 75)(28 37 53 76)(29 38 54 77)(30 39 55 78)(31 40 56 79)(32 33 49 80)

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

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

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

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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