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## G = C16×F5order 320 = 26·5

### Direct product of C16 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C16×F5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — C8×F5 — C16×F5
 Lower central C5 — C16×F5
 Upper central C1 — C16

Generators and relations for C16×F5
G = < a,b,c | a16=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C16×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 44 80 54 30)(2 45 65 55 31)(3 46 66 56 32)(4 47 67 57 17)(5 48 68 58 18)(6 33 69 59 19)(7 34 70 60 20)(8 35 71 61 21)(9 36 72 62 22)(10 37 73 63 23)(11 38 74 64 24)(12 39 75 49 25)(13 40 76 50 26)(14 41 77 51 27)(15 42 78 52 28)(16 43 79 53 29)
(17 57 47 67)(18 58 48 68)(19 59 33 69)(20 60 34 70)(21 61 35 71)(22 62 36 72)(23 63 37 73)(24 64 38 74)(25 49 39 75)(26 50 40 76)(27 51 41 77)(28 52 42 78)(29 53 43 79)(30 54 44 80)(31 55 45 65)(32 56 46 66)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4L 5 8A 8B 8C 8D 8E ··· 8P 10 16A ··· 16H 16I ··· 16AF 20A 20B 40A 40B 40C 40D 80A ··· 80H order 1 2 2 2 4 4 4 ··· 4 5 8 8 8 8 8 ··· 8 10 16 ··· 16 16 ··· 16 20 20 40 40 40 40 80 ··· 80 size 1 1 5 5 1 1 5 ··· 5 4 1 1 1 1 5 ··· 5 4 1 ··· 1 5 ··· 5 4 4 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C4 C8 C8 C16 F5 C2×F5 C4×F5 C8×F5 C16×F5 kernel C16×F5 D5×C16 D5⋊C16 C8×F5 C5⋊2C16 C80 C5⋊C16 D5⋊C8 C4×F5 C5⋊C8 C2×F5 F5 C16 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 2 2 8 8 32 1 1 2 4 8

Matrix representation of C16×F5 in GL5(𝔽241)

 76 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 0 0 240 0 1 0 0 240 0 0 1 0 240 0 0 0 1 240
,
 64 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(241))| [76,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

C16×F5 in GAP, Magma, Sage, TeX

C_{16}\times F_5
% in TeX

G:=Group("C16xF5");
// GroupNames label

G:=SmallGroup(320,181);
// by ID

G=gap.SmallGroup(320,181);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,80,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^16=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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