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G = C3≀C3.S3  order 486 = 2·35

The non-split extension by C3≀C3 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C3≀C3.S3, C9○He3⋊1S3, C9.(C32⋊C6), (C32×C9)⋊17C6, He3.C3⋊4S3, C9.He3⋊1C2, He3⋊C3⋊4S3, C32⋊4D9⋊8C3, He3.3(C3⋊S3), C33.67(C3×S3), C3.9(He3⋊4S3), (C3×C9).35(C3×S3), C32.19(C3×C3⋊S3), SmallGroup(486,175)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3≀C3.S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=b, ab=ba, cac-1=ab-1, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=ab-1c, ce=ec, cf=fc, de=ed, fdf=d-1, fef=b-1e2 >

Subgroups: 980 in 86 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C33⋊C2, C3≀C3, C3≀C3, He3.C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C9○He3, C33⋊C6, He3.S3, He3.2S3, He3.4S3, C32⋊4D9, C9.He3, C3≀C3.S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3⋊4S3, C3≀C3.S3

Character table of C3≀C3.S3

 class 123A3B3C3D3E3F3G3H3I6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O9P9Q
 size 18126666991818818122266666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-1111111111-1-111111111111111111    linear of order 2
ρ31-111111ζ32ζ3ζ32ζ3ζ6ζ6511111111111ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 6
ρ41-111111ζ3ζ32ζ3ζ32ζ65ζ611111111111ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 6
ρ51111111ζ32ζ3ζ32ζ3ζ32ζ311111111111ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ61111111ζ3ζ32ζ3ζ32ζ3ζ3211111111111ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ7202222222-1-100-1-1-1-1-1-1-1-1-1-1-12-1-1-12-1    orthogonal lifted from S3
ρ82022-1-1-122-1-100-1-1-1-1-1-1222-1-1-1-122-1-1    orthogonal lifted from S3
ρ92022-1-1-122-1-100222-122-1-1-1-1-1-12-1-1-12    orthogonal lifted from S3
ρ102022-1-1-1222200-1-1-12-1-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ112022-1-1-1-1+√-3-1-√-3-1+√-3-1-√-300-1-1-12-1-1-1-1-122ζ65ζ6ζ65ζ6ζ6ζ65    complex lifted from C3×S3
ρ122022-1-1-1-1-√-3-1+√-3-1-√-3-1+√-300-1-1-12-1-1-1-1-122ζ6ζ65ζ6ζ65ζ65ζ6    complex lifted from C3×S3
ρ132022222-1-√-3-1+√-3ζ6ζ6500-1-1-1-1-1-1-1-1-1-1-1-1-√-3ζ65ζ6ζ65-1+√-3ζ6    complex lifted from C3×S3
ρ142022-1-1-1-1+√-3-1-√-3ζ65ζ600-1-1-1-1-1-1222-1-1ζ65ζ6-1+√-3-1-√-3ζ6ζ65    complex lifted from C3×S3
ρ152022-1-1-1-1-√-3-1+√-3ζ6ζ6500222-122-1-1-1-1-1ζ6-1+√-3ζ6ζ65ζ65-1-√-3    complex lifted from C3×S3
ρ162022-1-1-1-1+√-3-1-√-3ζ65ζ600222-122-1-1-1-1-1ζ65-1-√-3ζ65ζ6ζ6-1+√-3    complex lifted from C3×S3
ρ172022222-1+√-3-1-√-3ζ65ζ600-1-1-1-1-1-1-1-1-1-1-1-1+√-3ζ6ζ65ζ6-1-√-3ζ65    complex lifted from C3×S3
ρ182022-1-1-1-1-√-3-1+√-3ζ6ζ6500-1-1-1-1-1-1222-1-1ζ6ζ65-1-√-3-1+√-3ζ65ζ6    complex lifted from C3×S3
ρ19606-30000000006660-3-300000000000    orthogonal lifted from C32⋊C6
ρ20606-3000000000-3-3-30-3600000000000    orthogonal lifted from C32⋊C6
ρ21606-3000000000-3-3-306-300000000000    orthogonal lifted from C32⋊C6
ρ2260-30-3030000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ92ζ95+2ζ94-ζ92+ζ900ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal faithful
ρ2360-3003-30000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ922ζ98-ζ94+ζ92+ζ900ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal faithful
ρ2460-303-300000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ94ζ95+2ζ94-ζ92+ζ900ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ922ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal faithful
ρ2560-303-300000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ92ζ98-ζ94+ζ92+ζ900ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal faithful
ρ2660-3003-30000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ9ζ95+2ζ94-ζ92+ζ9002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal faithful
ρ2760-303-300000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ92-ζ98+2ζ97+ζ94+ζ92002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ9ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal faithful
ρ2860-30-3030000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ9-ζ98+2ζ97+ζ94+ζ9200ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal faithful
ρ2960-30-3030000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ942ζ98-ζ94+ζ92+ζ9002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal faithful
ρ3060-3003-30000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ94-ζ98+2ζ97+ζ94+ζ9200ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal faithful

Permutation representations of C3≀C3.S3
►On 27 points - transitive group 27T170
Generators in S27
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)

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

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

(19 22 25)(20 23 26)(21 24 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)

(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 25)(20 24)(21 23)(26 27)
 
G:=sub<Sym(27)| (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,22,13)(2,23,14)(3,24,15)(4,25,16)(5,26,17)(6,27,18)(7,19,10)(8,20,11)(9,21,12), (19,22,25)(20,23,26)(21,24,27), (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), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,25)(20,24)(21,23)(26,27)>;
 
G:=Group( (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,22,13)(2,23,14)(3,24,15)(4,25,16)(5,26,17)(6,27,18)(7,19,10)(8,20,11)(9,21,12), (19,22,25)(20,23,26)(21,24,27), (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), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,25)(20,24)(21,23)(26,27) );
 
G=PermutationGroup([[(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(1,22,13),(2,23,14),(3,24,15),(4,25,16),(5,26,17),(6,27,18),(7,19,10),(8,20,11),(9,21,12)], [(19,22,25),(20,23,26),(21,24,27)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,16),(11,15),(12,14),(17,18),(19,25),(20,24),(21,23),(26,27)]])
 
G:=TransitiveGroup(27,170);
 

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

100000
010000
000100
00181800
00001818
000010
,
010000
18180000
000100
00181800
000001
00001818
,
001000
000100
000010
000001
100000
010000
,
100000
010000
001000
000100
000001
00001818
,
17120000
750000
00171200
007500
00001712
000075
,
100000
18180000
001000
00181800
000010
00001818

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

C3≀C3.S3 in GAP, Magma, Sage, TeX 

C_3\wr C_3.S_3
 
% in TeX
 
G:=Group("C3wrC3.S3");
 
// GroupNames label
 
G:=SmallGroup(486,175);
 
// by ID
 
G=gap.SmallGroup(486,175);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,4755,453,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=f^2=1,e^3=b,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,f*a*f=a^-1,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,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=b^-1*e^2>;
 
// generators/relations
 

Export

Character table of C3≀C3.S3 in TeX

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