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G = C9⋊C54order 486 = 2·35

The semidirect product of C9 and C54 acting via C54/C9=C6

metacyclic, supersoluble, monomial

Aliases: C9⋊C54, D9⋊C27, C92.C6, C9⋊C27⋊C2, (C3×D9).C9, (C9×D9).C3, C9.5(C9⋊C6), C3.3(S3×C27), (C3×C9).1C18, (C3×C27).1S3, C3.3(C9⋊C18), C32.14(S3×C9), (C3×C9).49(C3×S3), SmallGroup(486,30)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C54
C1C3C9C3×C9C92C9⋊C27 — C9⋊C54
C9 — C9⋊C54
C1C9

Generators and relations for C9⋊C54
 G = < a,b | a9=b54=1, bab-1=a2 >

9C2
2C3
3S3
9C6
2C9
2C9
6C9
3C3×S3
9C18
2C3×C9
3C27
6C27
3S3×C9
9C54
2C3×C27
3S3×C27

Smallest permutation representation of C9⋊C54
On 54 points
Generators in S54
(1 49 43 37 31 25 19 13 7)(2 44 32 20 8 50 38 26 14)(3 33 9 39 15 45 21 51 27)(4 10 16 22 28 34 40 46 52)(5 17 29 41 53 11 23 35 47)(6 30 54 24 48 18 42 12 36)
(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 46 47 48 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,49,43,37,31,25,19,13,7)(2,44,32,20,8,50,38,26,14)(3,33,9,39,15,45,21,51,27)(4,10,16,22,28,34,40,46,52)(5,17,29,41,53,11,23,35,47)(6,30,54,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,49,43,37,31,25,19,13,7)(2,44,32,20,8,50,38,26,14)(3,33,9,39,15,45,21,51,27)(4,10,16,22,28,34,40,46,52)(5,17,29,41,53,11,23,35,47)(6,30,54,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(1,49,43,37,31,25,19,13,7),(2,44,32,20,8,50,38,26,14),(3,33,9,39,15,45,21,51,27),(4,10,16,22,28,34,40,46,52),(5,17,29,41,53,11,23,35,47),(6,30,54,24,48,18,42,12,36)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])

90 conjugacy classes

class 1  2 3A3B3C3D3E6A6B9A···9F9G···9L9M···9U18A···18F27A···27R27S···27AJ54A···54R
order1233333669···99···99···918···1827···2727···2754···54
size1911222991···12···26···69···93···36···69···9

90 irreducible representations

dim111111112222666
type++++
imageC1C2C3C6C9C18C27C54S3C3×S3S3×C9S3×C27C9⋊C6C9⋊C18C9⋊C54
kernelC9⋊C54C9⋊C27C9×D9C92C3×D9C3×C9D9C9C3×C27C3×C9C32C3C9C3C1
# reps112266181812618126

Matrix representation of C9⋊C54 in GL6(𝔽109)

7500000
0380000
00105000
0001600
0000660
0000027
,
0000027
00010500
00001050
0027000
10500000
01050000

G:=sub<GL(6,GF(109))| [75,0,0,0,0,0,0,38,0,0,0,0,0,0,105,0,0,0,0,0,0,16,0,0,0,0,0,0,66,0,0,0,0,0,0,27],[0,0,0,0,105,0,0,0,0,0,0,105,0,0,0,27,0,0,0,105,0,0,0,0,0,0,105,0,0,0,27,0,0,0,0,0] >;

C9⋊C54 in GAP, Magma, Sage, TeX

C_9\rtimes C_{54}
% in TeX

G:=Group("C9:C54");
// GroupNames label

G:=SmallGroup(486,30);
// by ID

G=gap.SmallGroup(486,30);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,8104,3250,208,11669]);
// Polycyclic

G:=Group<a,b|a^9=b^54=1,b*a*b^-1=a^2>;
// generators/relations

Export

Subgroup lattice of C9⋊C54 in TeX

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