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G = C32⋊C9.S3order 486 = 2·35

1st non-split extension by C32⋊C9 of S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C32⋊C9.1S3, C32⋊C9.1C6, C33.2(C3×S3), C322D91C3, C32.24He32C2, C32.26(C32⋊C6), C3.5(He3.2S3), C3.1(He3.2C6), SmallGroup(486,5)

Series: Derived Chief Lower central Upper central

C1C3C32⋊C9 — C32⋊C9.S3
C1C3C32C33C32⋊C9C32.24He3 — C32⋊C9.S3
C32⋊C9 — C32⋊C9.S3
C1C3

Generators and relations for C32⋊C9.S3
 G = < a,b,c,d | a3=b3=c9=1, d6=b, ab=ba, cac-1=ab-1, dad-1=a-1bc3, bc=cb, bd=db, dcd-1=a-1c5 >

27C2
2C3
9C3
18C3
18C3
18C3
9S3
27C6
27S3
3C32
3C32
6C32
6C32
6C32
6C32
9C9
9C9
18C32
18C32
18C32
3C3⋊S3
9D9
9C3×S3
27C18
27C3×S3
3C3×C9
3C3×C9
6He3
6He3
6He3
6C33
3C3×C3⋊S3
9S3×C9
9C3×D9
2C3×He3
3C32⋊C18

Permutation representations of C32⋊C9.S3
On 18 points - transitive group 18T170
Generators in S18
(1 13 7)(4 16 10)(5 11 17)(6 12 18)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)
(1 17 9 7 5 15 13 11 3)(2 4 6 14 16 18 8 10 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)

G:=sub<Sym(18)| (1,13,7)(4,16,10)(5,11,17)(6,12,18), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18), (1,17,9,7,5,15,13,11,3)(2,4,6,14,16,18,8,10,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)>;

G:=Group( (1,13,7)(4,16,10)(5,11,17)(6,12,18), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18), (1,17,9,7,5,15,13,11,3)(2,4,6,14,16,18,8,10,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18) );

G=PermutationGroup([(1,13,7),(4,16,10),(5,11,17),(6,12,18)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18)], [(1,17,9,7,5,15,13,11,3),(2,4,6,14,16,18,8,10,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)])

G:=TransitiveGroup(18,170);

31 conjugacy classes

class 1  2 3A3B3C3D3E3F···3L6A6B9A···9F9G9H9I18A···18F
order12333333···3669···999918···18
size1271122218···1827279···918181827···27

31 irreducible representations

dim1111223666
type+++++
imageC1C2C3C6S3C3×S3He3.2C6C32⋊C6He3.2S3C32⋊C9.S3
kernelC32⋊C9.S3C32.24He3C322D9C32⋊C9C32⋊C9C33C3C32C3C1
# reps11221212136

Matrix representation of C32⋊C9.S3 in GL6(𝔽19)

700000
710000
8011000
000700
11000110
1200001
,
1100000
0110000
0011000
0001100
0000110
0000011
,
7130000
0127000
7120000
1270001
11701100
11700110
,
8000150
800087
800780
01100110
11070110
11000110

G:=sub<GL(6,GF(19))| [7,7,8,0,11,12,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,7,12,11,11,13,12,12,7,7,7,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[8,8,8,0,11,11,0,0,0,11,0,0,0,0,0,0,7,0,0,0,7,0,0,0,15,8,8,11,11,11,0,7,0,0,0,0] >;

C32⋊C9.S3 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_9.S_3
% in TeX

G:=Group("C3^2:C9.S3");
// GroupNames label

G:=SmallGroup(486,5);
// by ID

G=gap.SmallGroup(486,5);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,4755,873,735,3244]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^9=1,d^6=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b*c^3,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^5>;
// generators/relations

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Subgroup lattice of C32⋊C9.S3 in TeX

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