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## G = C9×D5order 90 = 2·32·5

### Direct product of C9 and D5

Aliases: C9×D5, C5⋊C18, C453C2, C15.C6, C3.(C3×D5), (C3×D5).C3, SmallGroup(90,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C9×D5
 Chief series C1 — C5 — C15 — C45 — C9×D5
 Lower central C5 — C9×D5
 Upper central C1 — C9

Generators and relations for C9×D5
G = < a,b,c | a9=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C9×D5
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 40 35 27 15)(2 41 36 19 16)(3 42 28 20 17)(4 43 29 21 18)(5 44 30 22 10)(6 45 31 23 11)(7 37 32 24 12)(8 38 33 25 13)(9 39 34 26 14)
(1 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)(19 41)(20 42)(21 43)(22 44)(23 45)(24 37)(25 38)(26 39)(27 40)

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

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

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

C9×D5 is a maximal subgroup of   C9⋊F5

36 conjugacy classes

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

36 irreducible representations

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

Matrix representation of C9×D5 in GL2(𝔽19) generated by

 4 0 0 4
,
 18 1 13 5
,
 5 15 6 14
G:=sub<GL(2,GF(19))| [4,0,0,4],[18,13,1,5],[5,6,15,14] >;

C9×D5 in GAP, Magma, Sage, TeX

C_9\times D_5
% in TeX

G:=Group("C9xD5");
// GroupNames label

G:=SmallGroup(90,2);
// by ID

G=gap.SmallGroup(90,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-5,29,1155]);
// Polycyclic

G:=Group<a,b,c|a^9=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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