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G = C3≀C3⋊S3  order 486 = 2·35

2nd semidirect product of C3≀C3 and S3 acting via S3/C3=C2

non-abelian, supersoluble, monomial

Aliases: C3≀C3⋊2S3, C9○He3⋊2S3, (C32×C9)⋊21S3, He3.C3⋊2S3, C9.He3⋊2C2, He3⋊C3⋊6S3, He3.5(C3⋊S3), C3.He3⋊2S3, C33.41(C3⋊S3), C9.5(He3⋊C2), C3.7(He3⋊5S3), C32.5(C33⋊C2), 3- 1+2.2(C3⋊S3), (C3×C9).14(C3⋊S3), SmallGroup(486,189)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C9.He3 — C3≀C3⋊S3
Chief series
C1 — C3 — C32 — C33 — C3≀C3 — C9.He3 — C3≀C3⋊S3
Lower central
C9.He3 — C3≀C3⋊S3
Upper central
C1

Generators and relations for C3≀C3⋊S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, eae-1=ab=ba, cac-1=faf=ab-1, ede-1=ad=da, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=ab-1c, ece-1=a-1c, fcf=a-1c-1, fdf=d-1, fef=e-1 >

Subgroups: 1060 in 119 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C3≀C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C33⋊S3, He3.3S3, He3⋊S3, 3- 1+2.S3, C3×C9⋊S3, He3.4S3, C9.He3, C3≀C3⋊S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, He3⋊5S3, C3≀C3⋊S3

Character table of C3≀C3⋊S3

 class 123A3B3C3D3E3F3G3H3I6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O9P9Q
 size 181233666181818818122266666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-1111111111-1-111111111111111111    linear of order 2
ρ320222-1-1-1-1-1-100-1-1-1-12-1-1-1-122-1222-1-1    orthogonal lifted from S3
ρ420222-1-1-1-12-100-1-1-1-1-1222-1-1-1-1-1-122-1    orthogonal lifted from S3
ρ520222-1-1-1-12-1002222-1-1-1-12-1-1-12-1-1-12    orthogonal lifted from S3
ρ620222-1-1-12-1-1002222-1-1-1-12-1-1-1-12-12-1    orthogonal lifted from S3
ρ720222-1-1-1-1-12002222-1-1-1-12-1-12-1-12-1-1    orthogonal lifted from S3
ρ820222222-1-1-10022222222222-1-1-1-1-1-1    orthogonal lifted from S3
ρ920222-1-1-12-1-100-1-1-1-1-1222-1-1-122-1-1-1-1    orthogonal lifted from S3
ρ1020222222-12-100-1-1-1-1-1-1-1-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ11202222222-1-100-1-1-1-1-1-1-1-1-1-1-1-1-1-12-12    orthogonal lifted from S3
ρ1220222-1-1-1-1-1-100-1-1-1-12-1-1-1-1222-1-1-122    orthogonal lifted from S3
ρ1320222222-1-1200-1-1-1-1-1-1-1-1-1-1-1-12-1-12-1    orthogonal lifted from S3
ρ1420222-1-1-122200-1-1-1-12-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ1520222-1-1-1-1-1200-1-1-1-1-1222-1-1-1-1-12-1-12    orthogonal lifted from S3
ρ163-13-3+3√-3/2-3-3√-3/2000000ζ6ζ65333-3+3√-3/20000-3-3√-3/200000000    complex lifted from He3⋊C2
ρ173-13-3-3√-3/2-3+3√-3/2000000ζ65ζ6333-3-3√-3/20000-3+3√-3/200000000    complex lifted from He3⋊C2
ρ18313-3+3√-3/2-3-3√-3/2000000ζ32ζ3333-3+3√-3/20000-3-3√-3/200000000    complex lifted from He3⋊C2
ρ19313-3-3√-3/2-3+3√-3/2000000ζ3ζ32333-3-3√-3/20000-3+3√-3/200000000    complex lifted from He3⋊C2
ρ2060-3003-30000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940ζ95+2ζ94-ζ92+ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ9202ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal faithful
ρ2160-30003-3000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940-ζ98+2ζ97+ζ94+ζ92ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ90ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal faithful
ρ2260-300-303000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90-ζ98+2ζ97+ζ94+ζ922ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ920ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal faithful
ρ2360-300-303000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ9402ζ98-ζ94+ζ92+ζ9ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ90-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal faithful
ρ2460-30003-3000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ9202ζ98-ζ94+ζ92+ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ920-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal faithful
ρ2560-30003-3000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90ζ95+2ζ94-ζ92+ζ9ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ902ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal faithful
ρ2660-3003-30000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920-ζ98+2ζ97+ζ94+ζ92ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ90ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal faithful
ρ2760-3003-30000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ902ζ98-ζ94+ζ92+ζ9ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ90-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal faithful
ρ2860-300-303000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920ζ95+2ζ94-ζ92+ζ9ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ902ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal faithful
ρ29606-3+3√-3-3-3√-300000000-3-3-33-3√-3/200003+3√-3/200000000    complex lifted from He3⋊5S3
ρ30606-3-3√-3-3+3√-300000000-3-3-33+3√-3/200003-3√-3/200000000    complex lifted from He3⋊5S3

Permutation representations of C3≀C3⋊S3
►On 27 points - transitive group 27T159
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

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

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

(1 3 2)(4 6 5)(7 8 9)(16 18 17)(22 24 23)(25 27 26)

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

(1 16)(2 18)(3 17)(4 22)(5 24)(6 23)(7 26)(8 27)(9 25)(10 13)(11 15)(12 14)(19 20)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,2,3)(4,5,6)(7,9,8)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,26,27), (1,14,22)(2,13,23)(3,15,24)(4,11,16)(5,10,17)(6,12,18)(7,21,25)(8,19,27)(9,20,26), (1,3,2)(4,6,5)(7,8,9)(16,18,17)(22,24,23)(25,27,26), (1,26,11)(2,27,10)(3,25,12)(4,22,21)(5,23,20)(6,24,19)(7,16,15)(8,18,13)(9,17,14), (1,16)(2,18)(3,17)(4,22)(5,24)(6,23)(7,26)(8,27)(9,25)(10,13)(11,15)(12,14)(19,20)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,2,3)(4,5,6)(7,9,8)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,26,27), (1,14,22)(2,13,23)(3,15,24)(4,11,16)(5,10,17)(6,12,18)(7,21,25)(8,19,27)(9,20,26), (1,3,2)(4,6,5)(7,8,9)(16,18,17)(22,24,23)(25,27,26), (1,26,11)(2,27,10)(3,25,12)(4,22,21)(5,23,20)(6,24,19)(7,16,15)(8,18,13)(9,17,14), (1,16)(2,18)(3,17)(4,22)(5,24)(6,23)(7,26)(8,27)(9,25)(10,13)(11,15)(12,14)(19,20) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,2,3),(4,5,6),(7,9,8),(10,12,11),(13,15,14),(16,17,18),(19,21,20),(22,23,24),(25,26,27)], [(1,14,22),(2,13,23),(3,15,24),(4,11,16),(5,10,17),(6,12,18),(7,21,25),(8,19,27),(9,20,26)], [(1,3,2),(4,6,5),(7,8,9),(16,18,17),(22,24,23),(25,27,26)], [(1,26,11),(2,27,10),(3,25,12),(4,22,21),(5,23,20),(6,24,19),(7,16,15),(8,18,13),(9,17,14)], [(1,16),(2,18),(3,17),(4,22),(5,24),(6,23),(7,26),(8,27),(9,25),(10,13),(11,15),(12,14),(19,20)]])
 
G:=TransitiveGroup(27,159);
 

Matrix representation of C3≀C3⋊S3 ►in GL6(𝔽19)

010000
18180000
001000
000100
00001818
000010
,
18180000
100000
00181800
001000
00001818
000010
,
005700
00121700
000057
00001217
12170000
2140000
,
100000
010000
000100
00181800
000001
00001818
,
000010
000001
100000
010000
001000
000100
,
1750000
720000
0000175
000072
0017500
007200

G:=sub<GL(6,GF(19))| [0,18,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[0,0,0,0,12,2,0,0,0,0,17,14,5,12,0,0,0,0,7,17,0,0,0,0,0,0,5,12,0,0,0,0,7,17,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[17,7,0,0,0,0,5,2,0,0,0,0,0,0,0,0,17,7,0,0,0,0,5,2,0,0,17,7,0,0,0,0,5,2,0,0] >;
 

C3≀C3⋊S3 in GAP, Magma, Sage, TeX 

C_3\wr C_3\rtimes S_3
 
% in TeX
 
G:=Group("C3wrC3:S3");
 
// GroupNames label
 
G:=SmallGroup(486,189);
 
// by ID
 
G=gap.SmallGroup(486,189);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1993,1951,218,867,303,11344,1096,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,e*a*e^-1=a*b=b*a,c*a*c^-1=f*a*f=a*b^-1,e*d*e^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,d*c*d^-1=a*b^-1*c,e*c*e^-1=a^-1*c,f*c*f=a^-1*c^-1,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of C3≀C3⋊S3 in TeX

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