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G = C9⋊S3  order 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

Derived series
C1 — C3×C9 — C9⋊S3
Chief series
C1 — C3 — C32 — C3×C9 — C9⋊S3
Lower central
C3×C9 — C9⋊S3
Upper central
C1

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

C1
27C2
C3
C3
C3
C3
9S3
9S3
9S3
9S3
C9
C32
C9
C9
3D9
3C3⋊S3
3D9
3D9
C3×C9
C9⋊S3

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
ρ720-1-12-1ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9    orthogonal lifted from D9
ρ820-1-12-1ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92    orthogonal lifted from D9
ρ9202-1-1-1ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ1020-12-1-1ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92    orthogonal lifted from D9
ρ1120-12-1-1ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9    orthogonal lifted from D9
ρ12202-1-1-1ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ1320-1-12-1ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94    orthogonal lifted from D9
ρ14202-1-1-1ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ1520-12-1-1ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94    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 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 19)(9 10 20)

(2 9)(3 8)(4 7)(5 6)(10 22)(11 21)(12 20)(13 19)(14 27)(15 26)(16 25)(17 24)(18 23)
 
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), (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,19)(9,10,20), (2,9)(3,8)(4,7)(5,6)(10,22)(11,21)(12,20)(13,19)(14,27)(15,26)(16,25)(17,24)(18,23)>;
 
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), (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,19)(9,10,20), (2,9)(3,8)(4,7)(5,6)(10,22)(11,21)(12,20)(13,19)(14,27)(15,26)(16,25)(17,24)(18,23) );
 
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)], [(1,11,21),(2,12,22),(3,13,23),(4,14,24),(5,15,25),(6,16,26),(7,17,27),(8,18,19),(9,10,20)], [(2,9),(3,8),(4,7),(5,6),(10,22),(11,21),(12,20),(13,19),(14,27),(15,26),(16,25),(17,24),(18,23)]])
 
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  C32⋊4D9  C9⋊S4  C32.3S4  C3⋊D45  C3⋊D63
C9⋊S3 is a maximal quotient of
 C9⋊Dic3  C9⋊D9  C32⋊2D9  C27⋊S3  C32⋊4D9  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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