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## G = C5×C9⋊C6order 270 = 2·33·5

### Direct product of C5 and C9⋊C6

Aliases: C5×C9⋊C6, C9⋊C30, D9⋊C15, C452C6, 3- 1+2⋊C10, (C5×D9)⋊C3, C32.(C5×S3), C3.3(S3×C15), C15.7(C3×S3), (C3×C15).2S3, (C5×3- 1+2)⋊2C2, SmallGroup(270,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C5×C9⋊C6
 Chief series C1 — C3 — C9 — C45 — C5×3- 1+2 — C5×C9⋊C6
 Lower central C9 — C5×C9⋊C6
 Upper central C1 — C5

Generators and relations for C5×C9⋊C6
G = < a,b,c | a5=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

Smallest permutation representation of C5×C9⋊C6
On 45 points
Generators in S45
(1 44 35 26 17)(2 45 36 27 18)(3 37 28 19 10)(4 38 29 20 11)(5 39 30 21 12)(6 40 31 22 13)(7 41 32 23 14)(8 42 33 24 15)(9 43 34 25 16)
(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)
(2 6 8 9 5 3)(4 7)(10 18 13 15 16 12)(11 14)(19 27 22 24 25 21)(20 23)(28 36 31 33 34 30)(29 32)(37 45 40 42 43 39)(38 41)

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

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

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

50 conjugacy classes

 class 1 2 3A 3B 3C 5A 5B 5C 5D 6A 6B 9A 9B 9C 10A 10B 10C 10D 15A 15B 15C 15D 15E ··· 15L 30A ··· 30H 45A ··· 45L order 1 2 3 3 3 5 5 5 5 6 6 9 9 9 10 10 10 10 15 15 15 15 15 ··· 15 30 ··· 30 45 ··· 45 size 1 9 2 3 3 1 1 1 1 9 9 6 6 6 9 9 9 9 2 2 2 2 3 ··· 3 9 ··· 9 6 ··· 6

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 S3 C3×S3 C5×S3 S3×C15 C9⋊C6 C5×C9⋊C6 kernel C5×C9⋊C6 C5×3- 1+2 C5×D9 C9⋊C6 C45 3- 1+2 D9 C9 C3×C15 C15 C32 C3 C5 C1 # reps 1 1 2 4 2 4 8 8 1 2 4 8 1 4

Matrix representation of C5×C9⋊C6 in GL6(𝔽181)

 135 0 0 0 0 0 0 135 0 0 0 0 0 0 135 0 0 0 0 0 0 135 0 0 0 0 0 0 135 0 0 0 0 0 0 135
,
 0 0 0 180 0 0 0 0 1 180 0 0 0 0 0 0 0 180 0 0 0 0 1 180 180 1 0 0 0 0 180 0 0 0 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 180 0 0 0 0 0 180 0 0

G:=sub<GL(6,GF(181))| [135,0,0,0,0,0,0,135,0,0,0,0,0,0,135,0,0,0,0,0,0,135,0,0,0,0,0,0,135,0,0,0,0,0,0,135],[0,0,0,0,180,180,0,0,0,0,1,0,0,1,0,0,0,0,180,180,0,0,0,0,0,0,0,1,0,0,0,0,180,180,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,180,180,0,0,0,1,0,0,0,0,1,0,0,0] >;

C5×C9⋊C6 in GAP, Magma, Sage, TeX

C_5\times C_9\rtimes C_6
% in TeX

G:=Group("C5xC9:C6");
// GroupNames label

G:=SmallGroup(270,11);
// by ID

G=gap.SmallGroup(270,11);
# by ID

G:=PCGroup([5,-2,-3,-5,-3,-3,3003,1208,138,4504]);
// Polycyclic

G:=Group<a,b,c|a^5=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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