C3^2:2D12 - GroupNames

G = C₃²⋊₂D₁₂ order 216 = 2³·3³

The semidirect product of C₃² and D₁₂ acting via D₁₂/C₃=D₄

Aliases: C₃³⋊₃D₄, C₃²⋊₂D₁₂, C₃²⋊C₄⋊S₃, C₃⋊₁S₃≀C₂, C₃⋊S₃.₂D₆, C₃²⋊₄D₆⋊₂C₂, (C₃×C₃²⋊C₄)⋊₁C₂, (C₃×C₃⋊S₃).₅C₂², SmallGroup(216,159)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃×C₃⋊S₃ — C₃²⋊₂D₁₂

Chief series

C₁ — C₃ — C₃³ — C₃×C₃⋊S₃ — C₃²⋊₄D₆ — C₃²⋊₂D₁₂

Lower central

C₃³ — C₃×C₃⋊S₃ — C₃²⋊₂D₁₂

Upper central

C₁

Generators and relations for C₃²⋊₂D₁₂
G = < a,b,c,d | a³=b³=c¹²=d²=1, ab=ba, cac^-1=b, dad=cbc^-1=a^-1, bd=db, dcd=c^-1 >

Subgroups: 436 in 60 conjugacy classes, 11 normal (9 characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², S₃, C₆, D₄, C₃², C₃², C₁₂, D₆, C₃×S₃, C₃⋊S₃, C₃⋊S₃, D₁₂, C₃³, C₃²⋊C₄, S₃², C₃×C₃⋊S₃, C₃×C₃⋊S₃, S₃≀C₂, C₃×C₃²⋊C₄, C₃²⋊₄D₆, C₃²⋊₂D₁₂
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, D₁₂, S₃≀C₂, C₃²⋊₂D₁₂

Character table of C₃²⋊₂D₁₂

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4	6A	6B	6C	12A	12B
size	1	9	18	18	2	4	4	8	8	18	18	36	36	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	-1	2	2	-1	-1	2	-1	0	0	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	0	0	-1	2	2	-1	-1	-2	-1	0	0	1	1	orthogonal lifted from D₆
ρ₇	2	-2	0	0	2	2	2	2	2	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₈	2	-2	0	0	-1	2	2	-1	-1	0	1	0	0	-√3	√3	orthogonal lifted from D₁₂
ρ₉	2	-2	0	0	-1	2	2	-1	-1	0	1	0	0	√3	-√3	orthogonal lifted from D₁₂
ρ₁₀	4	0	-2	0	4	-2	1	-2	1	0	0	0	1	0	0	orthogonal lifted from S₃≀C₂
ρ₁₁	4	0	0	2	4	1	-2	1	-2	0	0	-1	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₂	4	0	2	0	4	-2	1	-2	1	0	0	0	-1	0	0	orthogonal lifted from S₃≀C₂
ρ₁₃	4	0	0	-2	4	1	-2	1	-2	0	0	1	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₄	8	0	0	0	-4	-4	2	2	-1	0	0	0	0	0	0	orthogonal faithful
ρ₁₅	8	0	0	0	-4	2	-4	-1	2	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₃²⋊₂D₁₂
►On 12 points - transitive group 12T118

Generators in S₁₂

(2 6 10)(4 12 8)

(1 5 9)(3 11 7)

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

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

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

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

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

G:=TransitiveGroup(12,118);

►On 18 points - transitive group 18T104

Generators in S₁₈

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

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

(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 12)(8 11)(9 10)(13 18)(14 17)(15 16)

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

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

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

G:=TransitiveGroup(18,104);

►On 24 points - transitive group 24T558

Generators in S₂₄

(1 5 9)(3 11 7)(14 22 18)(16 20 24)

(2 10 6)(4 8 12)(13 21 17)(15 19 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)

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

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

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

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

G:=TransitiveGroup(24,558);

►On 27 points - transitive group 27T79

Generators in S₂₇

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

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

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

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

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

G:=TransitiveGroup(27,79);

C₃²⋊₂D₁₂ is a maximal subgroup of C₃³⋊₃SD₁₆ F₉⋊S₃ S₃×S₃≀C₂
C₃²⋊₂D₁₂ is a maximal quotient of (C₃×C₆).₈D₁₂ (C₃×C₆).₉D₁₂ C₃²⋊₂D₂₄ C₃³⋊₈SD₁₆ C₃³⋊₃Q₁₆

Polynomial with Galois group C₃²⋊₂D₁₂ over ℚ

action	f(x)	Disc(f)
12T118	x¹²-2x⁹-2x⁶+3x³-1	-3¹⁵·13⁶

Matrix representation of C₃²⋊₂D₁₂ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	12
0	0	0	1	0	0
0	0	0	0	1	0
0	0	1	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	0	12	0
0	0	0	1	12	0
0	0	0	0	0	1

7	10	0	0	0	0
3	10	0	0	0	0
0	0	0	12	0	0
0	0	0	0	0	12
0	0	12	0	0	0
0	0	0	0	12	0

7	10	0	0	0	0
3	6	0	0	0	0
0	0	0	0	0	12
0	0	0	12	0	0
0	0	0	0	12	0
0	0	12	0	0	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1],[7,3,0,0,0,0,10,10,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0,0,0] >;

C₃²⋊₂D₁₂ in GAP, Magma, Sage, TeX

C_3^2\rtimes_2D_{12}

% in TeX

G:=Group("C3^2:2D12");

// GroupNames label

G:=SmallGroup(216,159);

// by ID

G=gap.SmallGroup(216,159);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,31,579,585,111,244,130,376,5189]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^12=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;

// generators/relations

Export

Character table of C₃²⋊₂D₁₂ in TeX

G = C32⋊2D12 order 216 = 23·33

The semidirect product of C32 and D12 acting via D12/C3=D4

G = C₃²⋊₂D₁₂ order 216 = 2³·3³

The semidirect product of C₃² and D₁₂ acting via D₁₂/C₃=D₄