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## G = C9×F5order 180 = 22·32·5

### Direct product of C9 and F5

Aliases: C9×F5, C5⋊C36, C452C4, C15.C12, D5.C18, C3.(C3×F5), (C3×F5).C3, (C3×D5).2C6, (C9×D5).2C2, SmallGroup(180,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C9×F5
 Chief series C1 — C5 — C15 — C3×D5 — C9×D5 — C9×F5
 Lower central C5 — C9×F5
 Upper central C1 — C9

Generators and relations for C9×F5
G = < a,b,c | a9=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C9×F5
On 45 points
Generators in S45
(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)
(1 29 22 16 40)(2 30 23 17 41)(3 31 24 18 42)(4 32 25 10 43)(5 33 26 11 44)(6 34 27 12 45)(7 35 19 13 37)(8 36 20 14 38)(9 28 21 15 39)
(10 32 25 43)(11 33 26 44)(12 34 27 45)(13 35 19 37)(14 36 20 38)(15 28 21 39)(16 29 22 40)(17 30 23 41)(18 31 24 42)

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

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

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

45 conjugacy classes

 class 1 2 3A 3B 4A 4B 5 6A 6B 9A ··· 9F 12A 12B 12C 12D 15A 15B 18A ··· 18F 36A ··· 36L 45A ··· 45F order 1 2 3 3 4 4 5 6 6 9 ··· 9 12 12 12 12 15 15 18 ··· 18 36 ··· 36 45 ··· 45 size 1 5 1 1 5 5 4 5 5 1 ··· 1 5 5 5 5 4 4 5 ··· 5 5 ··· 5 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 type + + + image C1 C2 C3 C4 C6 C9 C12 C18 C36 F5 C3×F5 C9×F5 kernel C9×F5 C9×D5 C3×F5 C45 C3×D5 F5 C15 D5 C5 C9 C3 C1 # reps 1 1 2 2 2 6 4 6 12 1 2 6

Matrix representation of C9×F5 in GL5(𝔽181)

 39 0 0 0 0 0 132 0 0 0 0 0 132 0 0 0 0 0 132 0 0 0 0 0 132
,
 1 0 0 0 0 0 180 180 180 180 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 19 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 180 180 180 180

G:=sub<GL(5,GF(181))| [39,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132],[1,0,0,0,0,0,180,1,0,0,0,180,0,1,0,0,180,0,0,1,0,180,0,0,0],[19,0,0,0,0,0,1,0,0,180,0,0,0,1,180,0,0,0,0,180,0,0,1,0,180] >;

C9×F5 in GAP, Magma, Sage, TeX

C_9\times F_5
% in TeX

G:=Group("C9xF5");
// GroupNames label

G:=SmallGroup(180,5);
// by ID

G=gap.SmallGroup(180,5);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,66,1804,614]);
// Polycyclic

G:=Group<a,b,c|a^9=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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