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G = C5×C9⋊C6order 270 = 2·33·5

Direct product of C5 and C9⋊C6

direct product, metacyclic, supersoluble, monomial

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

C1C9 — C5×C9⋊C6
C1C3C9C45C5×3- 1+2 — C5×C9⋊C6
C9 — C5×C9⋊C6
C1C5

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

9C2
3C3
3S3
9C6
2C9
9C10
3C15
3C3×S3
3C5×S3
9C30
2C45
3S3×C15

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 3A3B3C5A5B5C5D6A6B9A9B9C10A10B10C10D15A15B15C15D15E···15L30A···30H45A···45L
order12333555566999101010101515151515···1530···3045···45
size19233111199666999922223···39···96···6

50 irreducible representations

dim11111111222266
type++++
imageC1C2C3C5C6C10C15C30S3C3×S3C5×S3S3×C15C9⋊C6C5×C9⋊C6
kernelC5×C9⋊C6C5×3- 1+2C5×D9C9⋊C6C453- 1+2D9C9C3×C15C15C32C3C5C1
# reps11242488124814

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

13500000
01350000
00135000
00013500
00001350
00000135
,
00018000
00118000
00000180
00001180
18010000
18000000
,
010000
100000
000001
000010
00118000
00018000

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

Export

Subgroup lattice of C5×C9⋊C6 in TeX

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