Copied to
clipboard

G = C5×C9⋊C6  order 270 = 2·33·5

Direct product of C5 and C9⋊C6

direct product, metacyclic, supersoluble, monomial

Aliases: C5×C9⋊C6, C9⋊C30, D9⋊C15, C45⋊2C6, 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 >

C1
9C2
C3
3C3
C5
3S3
9C6
C32
C9
2C9
9C10
C15
3C15
D9
3C3×S3
3- 1+2
3C5×S3
9C30
C45
C3×C15
2C45
C9⋊C6
C5×D9
3S3×C15
C5×3- 1+2
C5×C9⋊C6

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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁