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G = C9⋊S3order 54 = 2·33

The semidirect product of C9 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C9⋊S3, C3⋊D9, C32.3S3, (C3×C9)⋊3C2, C3.(C3⋊S3), SmallGroup(54,7)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C9⋊S3
C1C3C32C3×C9 — C9⋊S3
C3×C9 — C9⋊S3
C1

Generators and relations for C9⋊S3
 G = < a,b,c | a9=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

27C2
9S3
9S3
9S3
9S3
3D9
3C3⋊S3
3D9
3D9

Character table of C9⋊S3

 class 123A3B3C3D9A9B9C9D9E9F9G9H9I
 size 1272222222222222
ρ1111111111111111    trivial
ρ21-11111111111111    linear of order 2
ρ320-1-1-12-1-1222-1-1-1-1    orthogonal lifted from S3
ρ4202222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ520-1-1-12-1-1-1-1-1222-1    orthogonal lifted from S3
ρ620-1-1-1222-1-1-1-1-1-12    orthogonal lifted from S3
ρ720-1-12-1ζ9792ζ9594ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ989    orthogonal lifted from D9
ρ820-1-12-1ζ9594ζ989ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9792    orthogonal lifted from D9
ρ9202-1-1-1ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1020-12-1-1ζ9594ζ989ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ1120-12-1-1ζ9792ζ9594ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ989    orthogonal lifted from D9
ρ12202-1-1-1ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1320-1-12-1ζ989ζ9792ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ9594    orthogonal lifted from D9
ρ14202-1-1-1ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ1520-12-1-1ζ989ζ9792ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9594    orthogonal lifted from D9

Permutation representations of C9⋊S3
On 27 points - transitive group 27T10
Generators in S27
(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)
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 17 25)(6 18 26)(7 10 27)(8 11 19)(9 12 20)
(2 9)(3 8)(4 7)(5 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 27)(17 26)(18 25)

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

G:=Group( (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), (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,17,25)(6,18,26)(7,10,27)(8,11,19)(9,12,20), (2,9)(3,8)(4,7)(5,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,27)(17,26)(18,25) );

G=PermutationGroup([(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)], [(1,13,21),(2,14,22),(3,15,23),(4,16,24),(5,17,25),(6,18,26),(7,10,27),(8,11,19),(9,12,20)], [(2,9),(3,8),(4,7),(5,6),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,27),(17,26),(18,25)])

G:=TransitiveGroup(27,10);

C9⋊S3 is a maximal subgroup of
S3×D9  C32⋊D9  He3.S3  He3.2S3  C9⋊D9  C27⋊S3  C33.S3  He3.4S3  C324D9  C9⋊S4  C32.3S4  C3⋊D45  C3⋊D63
C9⋊S3 is a maximal quotient of
C9⋊Dic3  C9⋊D9  C322D9  C27⋊S3  C324D9  C9⋊S4  C32.3S4  C3⋊D45  C3⋊D63

Matrix representation of C9⋊S3 in GL4(𝔽19) generated by

121700
21400
00214
0057
,
1000
0100
001818
0010
,
1000
181800
001818
0001
G:=sub<GL(4,GF(19))| [12,2,0,0,17,14,0,0,0,0,2,5,0,0,14,7],[1,0,0,0,0,1,0,0,0,0,18,1,0,0,18,0],[1,18,0,0,0,18,0,0,0,0,18,0,0,0,18,1] >;

C9⋊S3 in GAP, Magma, Sage, TeX

C_9\rtimes S_3
% in TeX

G:=Group("C9:S3");
// GroupNames label

G:=SmallGroup(54,7);
// by ID

G=gap.SmallGroup(54,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,177,149,146,579]);
// Polycyclic

G:=Group<a,b,c|a^9=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9⋊S3 in TeX
Character table of C9⋊S3 in TeX

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