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G = C9⋊C36order 324 = 22·34

The semidirect product of C9 and C36 acting via C36/C6=C6

metacyclic, supersoluble, monomial

Aliases: C9⋊C36, C18.C18, Dic9⋊C9, C9⋊C9⋊C4, (C3×C9).C12, C2.(C9⋊C18), C6.3(S3×C9), C6.4(C9⋊C6), (C3×C18).4C6, (C3×C18).2S3, (C3×Dic9).C3, C3.2(C9⋊C12), C3.3(C9×Dic3), (C3×C9).1Dic3, C32.14(C3×Dic3), (C2×C9⋊C9).C2, (C3×C6).28(C3×S3), SmallGroup(324,9)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C36
C1C3C9C3×C9C3×C18C2×C9⋊C9 — C9⋊C36
C9 — C9⋊C36
C1C6

Generators and relations for C9⋊C36
 G = < a,b | a9=b36=1, bab-1=a5 >

2C3
9C4
2C6
2C9
3C9
6C9
3Dic3
9C12
2C18
3C18
6C18
2C3×C9
3C3×Dic3
9C36
2C3×C18
3C9×Dic3

Smallest permutation representation of C9⋊C36
On 36 points
Generators in S36
(1 9 29 25 33 17 13 21 5)(2 18 10 14 30 22 26 6 34)(3 23 19 27 11 7 15 35 31)(4 8 24 16 20 36 28 32 12)
(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)

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

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

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

60 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E9A···9F9G···9O12A12B12C12D18A···18F18G···18O36A···36L
order123333344666669···99···91212121218···1818···1836···36
size111122299112223···36···699993···36···69···9

60 irreducible representations

dim1111111112222226666
type+++-+-
imageC1C2C3C4C6C9C12C18C36S3Dic3C3×S3C3×Dic3S3×C9C9×Dic3C9⋊C6C9⋊C12C9⋊C18C9⋊C36
kernelC9⋊C36C2×C9⋊C9C3×Dic9C9⋊C9C3×C18Dic9C3×C9C18C9C3×C18C3×C9C3×C6C32C6C3C6C3C2C1
# reps11222646121122661122

Matrix representation of C9⋊C36 in GL6(𝔽37)

101127000
0026000
91127000
0001036
000163611
0000100
,
00010126
000122736
000010
273611000
25101000
0360000

G:=sub<GL(6,GF(37))| [10,0,9,0,0,0,11,0,11,0,0,0,27,26,27,0,0,0,0,0,0,1,16,0,0,0,0,0,36,10,0,0,0,36,11,0],[0,0,0,27,25,0,0,0,0,36,10,36,0,0,0,11,1,0,10,12,0,0,0,0,1,27,1,0,0,0,26,36,0,0,0,0] >;

C9⋊C36 in GAP, Magma, Sage, TeX

C_9\rtimes C_{36}
% in TeX

G:=Group("C9:C36");
// GroupNames label

G:=SmallGroup(324,9);
// by ID

G=gap.SmallGroup(324,9);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,5404,2170,208,7781]);
// Polycyclic

G:=Group<a,b|a^9=b^36=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C9⋊C36 in TeX

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