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G = C2×C33⋊C6  order 324 = 22·34

Direct product of C2 and C33⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C33⋊C6, He3⋊2D6, C3≀C3⋊4C22, (C2×He3)⋊1S3, C33⋊2(C2×C6), (C32×C6)⋊1C6, C33⋊C2⋊2C6, C32.5(S3×C6), C6.6(C32⋊C6), (C2×C3≀C3)⋊3C2, (C3×C6).16(C3×S3), C3.2(C2×C32⋊C6), (C2×C33⋊C2)⋊1C3, SmallGroup(324,69)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C2×C33⋊C6
Chief series
C1 — C3 — C32 — C33 — C3≀C3 — C33⋊C6 — C2×C33⋊C6
Lower central
C33 — C2×C33⋊C6
Upper central
C1 — C2

Generators and relations for C2×C33⋊C6
 G = < a,b,c,d,e | a2=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1c-1, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 772 in 84 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, He3, 3- 1+2, C33, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, C2×3- 1+2, C33⋊C2, C32×C6, C3≀C3, C2×C32⋊C6, C2×C33⋊C2, C33⋊C6, C2×C3≀C3, C2×C33⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, C33⋊C6, C2×C33⋊C6

Character table of C2×C33⋊C6

 class 12A2B2C3A3B3C3D3E3F3G6A6B6C6D6E6F6G6H6I6J6K9A9B18A18B
 size 112727266669926666992727272718181818
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
ρ5111111111ζ3ζ3211111ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ32    linear of order 3
ρ611-1-111111ζ3ζ3211111ζ32ζ3ζ65ζ65ζ6ζ6ζ3ζ32ζ3ζ32    linear of order 6
ρ71-11-111111ζ32ζ3-1-1-1-1-1ζ65ζ6ζ32ζ6ζ3ζ65ζ32ζ3ζ6ζ65    linear of order 6
ρ81-1-1111111ζ32ζ3-1-1-1-1-1ζ65ζ6ζ6ζ32ζ65ζ3ζ32ζ3ζ6ζ65    linear of order 6
ρ9111111111ζ32ζ311111ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ101-11-111111ζ3ζ32-1-1-1-1-1ζ6ζ65ζ3ζ65ζ32ζ6ζ3ζ32ζ65ζ6    linear of order 6
ρ1111-1-111111ζ32ζ311111ζ3ζ32ζ6ζ6ζ65ζ65ζ32ζ3ζ32ζ3    linear of order 6
ρ121-1-1111111ζ3ζ32-1-1-1-1-1ζ6ζ65ζ65ζ3ζ6ζ32ζ3ζ32ζ65ζ6    linear of order 6
ρ132-2002-1-12-122-211-21-2-20000-1-111    orthogonal lifted from D6
ρ1422002-1-12-1222-1-12-1220000-1-1-1-1    orthogonal lifted from S3
ρ152-2002-1-12-1-1+√-3-1-√-3-211-211+√-31-√-30000ζ65ζ6ζ3ζ32    complex lifted from S3×C6
ρ1622002-1-12-1-1+√-3-1-√-32-1-12-1-1-√-3-1+√-30000ζ65ζ6ζ65ζ6    complex lifted from C3×S3
ρ172-2002-1-12-1-1-√-3-1+√-3-211-211-√-31+√-30000ζ6ζ65ζ32ζ3    complex lifted from S3×C6
ρ1822002-1-12-1-1-√-3-1+√-32-1-12-1-1+√-3-1-√-30000ζ6ζ65ζ6ζ65    complex lifted from C3×S3
ρ196-600-33-300003030-30000000000    orthogonal faithful
ρ206600600-3000600-300000000000    orthogonal lifted from C32⋊C6
ρ216600-33-30000-30-3030000000000    orthogonal lifted from C33⋊C6
ρ226600-3-300300-3300-30000000000    orthogonal lifted from C33⋊C6
ρ236600-3030-300-3-33000000000000    orthogonal lifted from C33⋊C6
ρ246-600-3-3003003-30030000000000    orthogonal faithful
ρ256-600600-3000-600300000000000    orthogonal lifted from C2×C32⋊C6
ρ266-600-3030-30033-3000000000000    orthogonal faithful

Permutation representations of C2×C33⋊C6
►On 18 points - transitive group 18T125
Generators in S18
(1 4)(2 5)(3 6)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)

(2 16 7)(3 8 17)(5 10 13)(6 14 11)

(1 12 15)(3 17 8)(4 18 9)(6 11 14)

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

(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
 
G:=sub<Sym(18)| (1,4)(2,5)(3,6)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (2,16,7)(3,8,17)(5,10,13)(6,14,11), (1,12,15)(3,17,8)(4,18,9)(6,11,14), (1,15,12)(2,7,16)(3,17,8)(4,9,18)(5,13,10)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
 
G:=Group( (1,4)(2,5)(3,6)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (2,16,7)(3,8,17)(5,10,13)(6,14,11), (1,12,15)(3,17,8)(4,18,9)(6,11,14), (1,15,12)(2,7,16)(3,17,8)(4,9,18)(5,13,10)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
 
G=PermutationGroup([[(1,4),(2,5),(3,6),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(2,16,7),(3,8,17),(5,10,13),(6,14,11)], [(1,12,15),(3,17,8),(4,18,9),(6,11,14)], [(1,15,12),(2,7,16),(3,17,8),(4,9,18),(5,13,10),(6,11,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])
 
G:=TransitiveGroup(18,125);
 

Matrix representation of C2×C33⋊C6 ►in GL6(ℤ)

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

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,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0] >;
 

C2×C33⋊C6 in GAP, Magma, Sage, TeX 

C_2\times C_3^3\rtimes C_6
 
% in TeX
 
G:=Group("C2xC3^3:C6");
 
// GroupNames label
 
G:=SmallGroup(324,69);
 
// by ID
 
G=gap.SmallGroup(324,69);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,303,2164,1096,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1*c^-1,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C2×C33⋊C6 in TeX

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