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G = C2×C33⋊S3  order 324 = 22·34

Direct product of C2 and C33⋊S3

direct product, non-abelian, supersoluble, monomial

Aliases: C2×C33⋊S3, He3⋊3D6, C33⋊8D6, 3- 1+2⋊1D6, C3≀C3⋊2C22, (C2×He3)⋊2S3, (C32×C6)⋊2S3, C6.7(He3⋊C2), (C2×3- 1+2)⋊1S3, (C2×C3≀C3)⋊1C2, (C3×C6).5(C3⋊S3), C32.1(C2×C3⋊S3), C3.2(C2×He3⋊C2), SmallGroup(324,77)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3≀C3 — C2×C33⋊S3
Chief series
C1 — C3 — C32 — C33 — C3≀C3 — C33⋊S3 — C2×C33⋊S3
Lower central
C3≀C3 — C2×C33⋊S3
Upper central
C1 — C2

Generators and relations for C2×C33⋊S3
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=b-1, ece-1=cd=dc, fcf=cd-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 634 in 100 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, D9, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, He3, 3- 1+2, C33, D18, S3×C6, C2×C3⋊S3, C32⋊C6, C9⋊C6, C2×He3, C2×3- 1+2, C3×C3⋊S3, C32×C6, C3≀C3, C2×C32⋊C6, C2×C9⋊C6, C6×C3⋊S3, C33⋊S3, C2×C3≀C3, C2×C33⋊S3
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, C33⋊S3, C2×C33⋊S3

Character table of C2×C33⋊S3

 class 12A2B2C3A3B3C3D3E3F3G6A6B6C6D6E6F6G6H6I6J6K9A9B18A18B
 size 11272723366618233666182727272718181818
ρ111111111111111111111111111    trivial
ρ21-1-111111111-1-1-1-1-1-1-1-11-1111-1-1    linear of order 2
ρ311-1-111111111111111-1-1-1-11111    linear of order 2
ρ41-11-11111111-1-1-1-1-1-1-11-11-111-1-1    linear of order 2
ρ52200222-1-1-12222-1-1-120000-1-1-1-1    orthogonal lifted from S3
ρ62-200222-1-1-12-2-2-2111-20000-1-111    orthogonal lifted from D6
ρ72-200222-1-1-1-1-2-2-2111100002-1-21    orthogonal lifted from D6
ρ82200222-1-1-1-1222-1-1-1-100002-12-1    orthogonal lifted from S3
ρ92-200222-1-1-1-1-2-2-211110000-121-2    orthogonal lifted from D6
ρ102200222-1-1-1-1222-1-1-1-10000-12-12    orthogonal lifted from S3
ρ112200222222-1222222-10000-1-1-1-1    orthogonal lifted from S3
ρ122-200222222-1-2-2-2-2-2-210000-1-111    orthogonal lifted from D6
ρ1333113-3-3√-3/2-3+3√-3/200003-3-3√-3/2-3+3√-3/20000ζ32ζ32ζ3ζ30000    complex lifted from He3⋊C2
ρ1433-1-13-3-3√-3/2-3+3√-3/200003-3-3√-3/2-3+3√-3/20000ζ6ζ6ζ65ζ650000    complex lifted from He3⋊C2
ρ153-31-13-3-3√-3/2-3+3√-3/20000-33+3√-3/23-3√-3/20000ζ32ζ6ζ3ζ650000    complex lifted from C2×He3⋊C2
ρ163-3-113-3-3√-3/2-3+3√-3/20000-33+3√-3/23-3√-3/20000ζ6ζ32ζ65ζ30000    complex lifted from C2×He3⋊C2
ρ1733-1-13-3+3√-3/2-3-3√-3/200003-3+3√-3/2-3-3√-3/20000ζ65ζ65ζ6ζ60000    complex lifted from He3⋊C2
ρ183-31-13-3+3√-3/2-3-3√-3/20000-33-3√-3/23+3√-3/20000ζ3ζ65ζ32ζ60000    complex lifted from C2×He3⋊C2
ρ1933113-3+3√-3/2-3-3√-3/200003-3+3√-3/2-3-3√-3/20000ζ3ζ3ζ32ζ320000    complex lifted from He3⋊C2
ρ203-3-113-3+3√-3/2-3-3√-3/20000-33-3√-3/23+3√-3/20000ζ65ζ3ζ6ζ320000    complex lifted from C2×He3⋊C2
ρ216-600-300-3030300-330000000000    orthogonal faithful
ρ226600-300-3030-3003-30000000000    orthogonal lifted from C33⋊S3
ρ236-600-30003-3030030-3000000000    orthogonal faithful
ρ246-600-3003-3003000-33000000000    orthogonal faithful
ρ256600-30003-30-300-303000000000    orthogonal lifted from C33⋊S3
ρ266600-3003-300-30003-3000000000    orthogonal lifted from C33⋊S3

Permutation representations of C2×C33⋊S3
►On 18 points - transitive group 18T134
Generators in S18
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 2 3)(7 8 9)(10 11 12)(16 17 18)

(1 3 2)(4 5 6)(7 8 9)(10 12 11)(13 14 15)(16 17 18)

(1 5 8)(2 4 7)(3 6 9)(10 14 17)(11 13 16)(12 15 18)

(1 10)(2 12)(3 11)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)
 
G:=sub<Sym(18)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,2,3)(7,8,9)(10,11,12)(16,17,18), (1,3,2)(4,5,6)(7,8,9)(10,12,11)(13,14,15)(16,17,18), (1,5,8)(2,4,7)(3,6,9)(10,14,17)(11,13,16)(12,15,18), (1,10)(2,12)(3,11)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)>;
 
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,2,3)(7,8,9)(10,11,12)(16,17,18), (1,3,2)(4,5,6)(7,8,9)(10,12,11)(13,14,15)(16,17,18), (1,5,8)(2,4,7)(3,6,9)(10,14,17)(11,13,16)(12,15,18), (1,10)(2,12)(3,11)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13) );
 
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,2,3),(7,8,9),(10,11,12),(16,17,18)], [(1,3,2),(4,5,6),(7,8,9),(10,12,11),(13,14,15),(16,17,18)], [(1,5,8),(2,4,7),(3,6,9),(10,14,17),(11,13,16),(12,15,18)], [(1,10),(2,12),(3,11),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13)]])
 
G:=TransitiveGroup(18,134);
 

Matrix representation of C2×C33⋊S3 ►in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
010000
-1-10000
001000
000100
000010
000001
,
-1-10000
100000
001000
000100
000001
0000-1-1
,
010000
-1-10000
000100
00-1-100
000001
0000-1-1
,
000010
000001
100000
010000
001000
000100
,
100000
-1-10000
000010
0000-1-1
001000
00-1-100

G:=sub<GL(6,Integers())| [-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,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0] >;
 

C2×C33⋊S3 in GAP, Magma, Sage, TeX 

C_2\times C_3^3\rtimes S_3
 
% in TeX
 
G:=Group("C2xC3^3:S3");
 
// GroupNames label
 
G:=SmallGroup(324,77);
 
// by ID
 
G=gap.SmallGroup(324,77);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,579,303,7564,1096,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=b^-1,e*c*e^-1=c*d=d*c,f*c*f=c*d^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of C2×C33⋊S3 in TeX

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