C3^2:4D6 - GroupNames

G = C₃²⋊₄D₆ order 108 = 2²·3³

The semidirect product of C₃² and D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, A-group

Aliases: C₃²⋊₄D₆, C₃³⋊₃C₂², C₃⋊₂S₃², C₃⋊S₃⋊₂S₃, (C₃×C₃⋊S₃)⋊₃C₂, SmallGroup(108,40)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃²⋊₄D₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×C₃⋊S₃ — C₃²⋊₄D₆

Lower central

C₃³ — C₃²⋊₄D₆

Upper central

C₁

Generators and relations for C₃²⋊₄D₆
G = < a,b,c,d | a³=b³=c⁶=d²=1, ab=ba, cac^-1=a^-1, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 224 in 54 conjugacy classes, 15 normal (3 characteristic)
C₁, C₂, C₃, C₃, C₂², S₃, C₆, C₃², C₃², D₆, C₃×S₃, C₃⋊S₃, C₃³, S₃², C₃×C₃⋊S₃, C₃²⋊₄D₆
Quotients: C₁, C₂, C₂², S₃, D₆, S₃², C₃²⋊₄D₆

Character table of C₃²⋊₄D₆

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	3G	3H	6A	6B	6C
size	1	9	9	9	2	2	2	4	4	4	4	4	18	18	18
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	1	1	1	1	1	1	1	1	-1	-1	linear of order 2
ρ₃	1	-1	1	-1	1	1	1	1	1	1	1	1	-1	1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	1	1	1	1	1	1	1	-1	-1	1	linear of order 2
ρ₅	2	2	0	0	2	2	-1	-1	-1	-1	-1	2	0	0	-1	orthogonal lifted from S₃
ρ₆	2	0	-2	0	-1	2	2	-1	-1	-1	2	-1	0	1	0	orthogonal lifted from D₆
ρ₇	2	-2	0	0	2	2	-1	-1	-1	-1	-1	2	0	0	1	orthogonal lifted from D₆
ρ₈	2	0	2	0	-1	2	2	-1	-1	-1	2	-1	0	-1	0	orthogonal lifted from S₃
ρ₉	2	0	0	2	2	-1	2	-1	-1	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₁₀	2	0	0	-2	2	-1	2	-1	-1	2	-1	-1	1	0	0	orthogonal lifted from D₆
ρ₁₁	4	0	0	0	4	-2	-2	1	1	-2	1	-2	0	0	0	orthogonal lifted from S₃²
ρ₁₂	4	0	0	0	-2	4	-2	1	1	1	-2	-2	0	0	0	orthogonal lifted from S₃²
ρ₁₃	4	0	0	0	-2	-2	4	1	1	-2	-2	1	0	0	0	orthogonal lifted from S₃²
ρ₁₄	4	0	0	0	-2	-2	-2	^-1-3√-3/₂	^-1+3√-3/₂	1	1	1	0	0	0	complex faithful
ρ₁₅	4	0	0	0	-2	-2	-2	^-1+3√-3/₂	^-1-3√-3/₂	1	1	1	0	0	0	complex faithful

Permutation representations of C₃²⋊₄D₆
►On 12 points - transitive group 12T71

Generators in S₁₂

(1 5 3)(2 4 6)(7 11 9)(8 10 12)

(1 5 3)(2 4 6)(7 9 11)(8 12 10)

(1 2 3 4 5 6)(7 8 9 10 11 12)

(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)

G:=sub<Sym(12)| (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11)>;

G:=Group( (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11) );

G=PermutationGroup([[(1,5,3),(2,4,6),(7,11,9),(8,10,12)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10),(2,9),(3,8),(4,7),(5,12),(6,11)]])

G:=TransitiveGroup(12,71);

►On 18 points - transitive group 18T53

Generators in S₁₈

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

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

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

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

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

G:=Group( (1,13,16)(2,17,14)(3,15,18)(4,12,9)(5,10,7)(6,8,11), (1,13,16)(2,17,14)(3,15,18)(4,9,12)(5,7,10)(6,11,8), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,14)(8,13)(9,18)(10,17)(11,16)(12,15) );

G=PermutationGroup([[(1,13,16),(2,17,14),(3,15,18),(4,12,9),(5,10,7),(6,8,11)], [(1,13,16),(2,17,14),(3,15,18),(4,9,12),(5,7,10),(6,11,8)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,14),(8,13),(9,18),(10,17),(11,16),(12,15)]])

G:=TransitiveGroup(18,53);

►On 27 points - transitive group 27T35

Generators in S₂₇

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

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

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

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

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

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

G:=TransitiveGroup(27,35);

C₃²⋊₄D₆ is a maximal subgroup of
C₃³⋊D₄ C₃²⋊₂D₁₂ S₃³ C₃²⋊D₁₈ He₃⋊₅D₆ C₃²⋊₅D₁₈ C₃³⋊A₄ C₃³⋊₁₇D₆ C₆²⋊₁₀D₆
C₃²⋊₄D₆ is a maximal quotient of
C₃³⋊₉(C₂×C₄) C₃³⋊₉D₄ C₃³⋊₅Q₈ C₃²⋊₅D₁₈ He₃⋊₆D₆ He₃.₆D₆ C₃³⋊₁₇D₆ C₆²⋊₁₀D₆

Polynomial with Galois group C₃²⋊₄D₆ over ℚ

action	f(x)	Disc(f)
12T71	x¹²-5x⁹+11x⁶-10x³+4	2¹⁰·3¹⁸·5⁶

Matrix representation of C₃²⋊₄D₆ ►in GL₄(𝔽₇) generated by

5	3	2	3
1	3	3	0
4	4	0	6
0	0	0	4

3	6	3	2
6	3	4	2
0	0	2	0
0	0	0	4

5	2	2	1
0	6	1	1
2	5	6	2
5	5	2	4

5	2	0	1
3	3	3	6
3	4	2	6
5	5	1	4

G:=sub<GL(4,GF(7))| [5,1,4,0,3,3,4,0,2,3,0,0,3,0,6,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[5,0,2,5,2,6,5,5,2,1,6,2,1,1,2,4],[5,3,3,5,2,3,4,5,0,3,2,1,1,6,6,4] >;

C₃²⋊₄D₆ in GAP, Magma, Sage, TeX

C_3^2\rtimes_4D_6

% in TeX

G:=Group("C3^2:4D6");

// GroupNames label

G:=SmallGroup(108,40);

// by ID

G=gap.SmallGroup(108,40);

# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,122,67,248,1804]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

Export

Character table of C₃²⋊₄D₆ in TeX

G = C32⋊4D6 order 108 = 22·33

The semidirect product of C32 and D6 acting via D6/C3=C22

G = C₃²⋊₄D₆ order 108 = 2²·3³

The semidirect product of C₃² and D₆ acting via D₆/C₃=C₂²