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## G = C5×F9order 360 = 23·32·5

### Direct product of C5 and F9

Aliases: C5×F9, C32⋊C40, C3⋊S3.C20, (C3×C15)⋊1C8, C32⋊C4.1C10, (C5×C3⋊S3).1C4, (C5×C32⋊C4).1C2, SmallGroup(360,123)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×F9
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C5×C32⋊C4 — C5×F9
 Lower central C32 — C5×F9
 Upper central C1 — C5

Generators and relations for C5×F9
G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of C5×F9
On 45 points
Generators in S45
(1 2 4 3 5)(6 21 43 26 37)(7 14 44 27 30)(8 15 45 28 31)(9 16 38 29 32)(10 17 39 22 33)(11 18 40 23 34)(12 19 41 24 35)(13 20 42 25 36)
(1 19 15)(2 41 45)(3 35 31)(4 24 28)(5 12 8)(6 7 9)(10 13 11)(14 16 21)(17 20 18)(22 25 23)(26 27 29)(30 32 37)(33 36 34)(38 43 44)(39 42 40)
(1 20 16)(2 42 38)(3 36 32)(4 25 29)(5 13 9)(6 12 11)(7 8 10)(14 15 17)(18 21 19)(22 27 28)(23 26 24)(30 31 33)(34 37 35)(39 44 45)(40 43 41)
(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)

G:=sub<Sym(45)| (1,2,4,3,5)(6,21,43,26,37)(7,14,44,27,30)(8,15,45,28,31)(9,16,38,29,32)(10,17,39,22,33)(11,18,40,23,34)(12,19,41,24,35)(13,20,42,25,36), (1,19,15)(2,41,45)(3,35,31)(4,24,28)(5,12,8)(6,7,9)(10,13,11)(14,16,21)(17,20,18)(22,25,23)(26,27,29)(30,32,37)(33,36,34)(38,43,44)(39,42,40), (1,20,16)(2,42,38)(3,36,32)(4,25,29)(5,13,9)(6,12,11)(7,8,10)(14,15,17)(18,21,19)(22,27,28)(23,26,24)(30,31,33)(34,37,35)(39,44,45)(40,43,41), (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)>;

G:=Group( (1,2,4,3,5)(6,21,43,26,37)(7,14,44,27,30)(8,15,45,28,31)(9,16,38,29,32)(10,17,39,22,33)(11,18,40,23,34)(12,19,41,24,35)(13,20,42,25,36), (1,19,15)(2,41,45)(3,35,31)(4,24,28)(5,12,8)(6,7,9)(10,13,11)(14,16,21)(17,20,18)(22,25,23)(26,27,29)(30,32,37)(33,36,34)(38,43,44)(39,42,40), (1,20,16)(2,42,38)(3,36,32)(4,25,29)(5,13,9)(6,12,11)(7,8,10)(14,15,17)(18,21,19)(22,27,28)(23,26,24)(30,31,33)(34,37,35)(39,44,45)(40,43,41), (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) );

G=PermutationGroup([[(1,2,4,3,5),(6,21,43,26,37),(7,14,44,27,30),(8,15,45,28,31),(9,16,38,29,32),(10,17,39,22,33),(11,18,40,23,34),(12,19,41,24,35),(13,20,42,25,36)], [(1,19,15),(2,41,45),(3,35,31),(4,24,28),(5,12,8),(6,7,9),(10,13,11),(14,16,21),(17,20,18),(22,25,23),(26,27,29),(30,32,37),(33,36,34),(38,43,44),(39,42,40)], [(1,20,16),(2,42,38),(3,36,32),(4,25,29),(5,13,9),(6,12,11),(7,8,10),(14,15,17),(18,21,19),(22,27,28),(23,26,24),(30,31,33),(34,37,35),(39,44,45),(40,43,41)], [(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)]])

45 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 15A 15B 15C 15D 20A ··· 20H 40A ··· 40P order 1 2 3 4 4 5 5 5 5 8 8 8 8 10 10 10 10 15 15 15 15 20 ··· 20 40 ··· 40 size 1 9 8 9 9 1 1 1 1 9 9 9 9 9 9 9 9 8 8 8 8 9 ··· 9 9 ··· 9

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 8 8 type + + + image C1 C2 C4 C5 C8 C10 C20 C40 F9 C5×F9 kernel C5×F9 C5×C32⋊C4 C5×C3⋊S3 F9 C3×C15 C32⋊C4 C3⋊S3 C32 C5 C1 # reps 1 1 2 4 4 4 8 16 1 4

Matrix representation of C5×F9 in GL8(𝔽241)

 91 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 91
,
 0 0 1 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 0 0 240 0 0 1 0 0 0 0 240 1 0 0 0 0 0 0 240 0 1 0 0 1 0 0 240 0 0 0 0 0 0 0 240 0 0 0 1 0 0 0 240 0 0 0
,
 0 0 0 0 0 1 0 240 0 0 0 0 0 0 1 240 0 1 0 0 0 0 0 240 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 240 0 0 0 0 1 0 0 240 0 0 1 0 0 0 0 240 0 0 0 1 0 0 0 240
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,GF(241))| [91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91],[0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,240,240,240,240,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,240,240,240,240],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0] >;

C5×F9 in GAP, Magma, Sage, TeX

C_5\times F_9
% in TeX

G:=Group("C5xF9");
// GroupNames label

G:=SmallGroup(360,123);
// by ID

G=gap.SmallGroup(360,123);
# by ID

G:=PCGroup([6,-2,-5,-2,-2,-3,3,60,50,3604,856,142,10085,1169,455]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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