C3^2:C12 - GroupNames

G = C₃²:C₁₂ order 108 = 2²·3³

The semidirect product of C₃² and C₁₂ acting via C₁₂/C₂=C₆

Aliases: C₃²:C₁₂, He₃:₂C₄, C₃²:₁Dic₃, (C₃xC₆).C₆, C₃:Dic₃:C₃, C₆.₂(C₃xS₃), (C₃xC₆).₁S₃, C₂.(C₃²:C₆), (C₂xHe₃).₁C₂, C₃.₂(C₃xDic₃), SmallGroup(108,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃²:C₁₂

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₂xHe₃ — C₃²:C₁₂

Lower central

C₃² — C₃²:C₁₂

Upper central

C₁ — C₂

Generators and relations for C₃²:C₁₂
G = < a,b,c | a³=b³=c¹²=1, ab=ba, cac^-1=a^-1b, cbc^-1=b^-1 >

Subgroups: 73 in 25 conjugacy classes, 12 normal (all characteristic)
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, Dic₃, C₁₂, C₃xS₃, C₃xDic₃, C₃²:C₆, C₃²:C₁₂

C₁

C₂

C₃

₃C₃

₆C₃

₉C₄

C₆

₃C₆

₆C₆

C₃²

₂C₃²

₃Dic₃

₉C₁₂

₉Dic₃

C₃xC₆

₂C₃xC₆

He₃

C₃:Dic₃

₃C₃xDic₃

C₂xHe₃

C₃²:C₁₂

Character table of C₃²:C₁₂

class	1	2	3A	3B	3C	3D	3E	3F	4A	4B	6A	6B	6C	6D	6E	6F	12A	12B	12C	12D
size	1	1	2	3	3	6	6	6	9	9	2	3	3	6	6	6	9	9	9	9
ρ₁	1	1	1	1	1	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	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	1	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₄	1	1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	1	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₅	1	1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	-1	-1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	linear of order 6
ρ₆	1	1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	-1	-1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₇	1	-1	1	1	1	1	1	1	-i	i	-1	-1	-1	-1	-1	-1	i	-i	i	-i	linear of order 4
ρ₈	1	-1	1	1	1	1	1	1	i	-i	-1	-1	-1	-1	-1	-1	-i	i	-i	i	linear of order 4
ρ₉	1	-1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	i	-i	-1	ζ₆	ζ₆⁵	-1	ζ₆⁵	ζ₆	ζ₄³ζ₃	ζ₄ζ₃	ζ₄³ζ₃²	ζ₄ζ₃²	linear of order 12
ρ₁₀	1	-1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	i	-i	-1	ζ₆⁵	ζ₆	-1	ζ₆	ζ₆⁵	ζ₄³ζ₃²	ζ₄ζ₃²	ζ₄³ζ₃	ζ₄ζ₃	linear of order 12
ρ₁₁	1	-1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	-i	i	-1	ζ₆⁵	ζ₆	-1	ζ₆	ζ₆⁵	ζ₄ζ₃²	ζ₄³ζ₃²	ζ₄ζ₃	ζ₄³ζ₃	linear of order 12
ρ₁₂	1	-1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	-i	i	-1	ζ₆	ζ₆⁵	-1	ζ₆⁵	ζ₆	ζ₄ζ₃	ζ₄³ζ₃	ζ₄ζ₃²	ζ₄³ζ₃²	linear of order 12
ρ₁₃	2	2	2	2	2	-1	-1	-1	0	0	2	2	2	-1	-1	-1	0	0	0	0	orthogonal lifted from S₃
ρ₁₄	2	-2	2	2	2	-1	-1	-1	0	0	-2	-2	-2	1	1	1	0	0	0	0	symplectic lifted from Dic₃, Schur index 2
ρ₁₅	2	-2	2	-1+√-3	-1-√-3	ζ₆⁵	-1	ζ₆	0	0	-2	1-√-3	1+√-3	1	ζ₃²	ζ₃	0	0	0	0	complex lifted from C₃xDic₃
ρ₁₆	2	-2	2	-1-√-3	-1+√-3	ζ₆	-1	ζ₆⁵	0	0	-2	1+√-3	1-√-3	1	ζ₃	ζ₃²	0	0	0	0	complex lifted from C₃xDic₃
ρ₁₇	2	2	2	-1+√-3	-1-√-3	ζ₆⁵	-1	ζ₆	0	0	2	-1+√-3	-1-√-3	-1	ζ₆	ζ₆⁵	0	0	0	0	complex lifted from C₃xS₃
ρ₁₈	2	2	2	-1-√-3	-1+√-3	ζ₆	-1	ζ₆⁵	0	0	2	-1-√-3	-1+√-3	-1	ζ₆⁵	ζ₆	0	0	0	0	complex lifted from C₃xS₃
ρ₁₉	6	6	-3	0	0	0	0	0	0	0	-3	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²:C₆
ρ₂₀	6	-6	-3	0	0	0	0	0	0	0	3	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₃²:C₁₂
►On 36 points

Generators in S₃₆

(1 22 5)(2 19 10)(3 32 28)(4 8 13)(6 31 35)(7 16 11)(9 26 34)(12 25 29)(14 30 18)(15 27 23)(17 21 33)(20 36 24)

(1 14 34)(2 35 15)(3 16 36)(4 25 17)(5 18 26)(6 27 19)(7 20 28)(8 29 21)(9 22 30)(10 31 23)(11 24 32)(12 33 13)

(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 28 29 30 31 32 33 34 35 36)

G:=sub<Sym(36)| (1,22,5)(2,19,10)(3,32,28)(4,8,13)(6,31,35)(7,16,11)(9,26,34)(12,25,29)(14,30,18)(15,27,23)(17,21,33)(20,36,24), (1,14,34)(2,35,15)(3,16,36)(4,25,17)(5,18,26)(6,27,19)(7,20,28)(8,29,21)(9,22,30)(10,31,23)(11,24,32)(12,33,13), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)>;

G:=Group( (1,22,5)(2,19,10)(3,32,28)(4,8,13)(6,31,35)(7,16,11)(9,26,34)(12,25,29)(14,30,18)(15,27,23)(17,21,33)(20,36,24), (1,14,34)(2,35,15)(3,16,36)(4,25,17)(5,18,26)(6,27,19)(7,20,28)(8,29,21)(9,22,30)(10,31,23)(11,24,32)(12,33,13), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36) );

G=PermutationGroup([[(1,22,5),(2,19,10),(3,32,28),(4,8,13),(6,31,35),(7,16,11),(9,26,34),(12,25,29),(14,30,18),(15,27,23),(17,21,33),(20,36,24)], [(1,14,34),(2,35,15),(3,16,36),(4,25,17),(5,18,26),(6,27,19),(7,20,28),(8,29,21),(9,22,30),(10,31,23),(11,24,32),(12,33,13)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)]])

C₃²:C₁₂ is a maximal subgroup of
He₃:₂Q₈ C₆.S₃² He₃:(C₂xC₄) He₃:₃D₄ He₃:₃Q₈ C₄xC₃²:C₆ He₃:₆D₄ C₃³:C₁₂ He₃.Dic₃ He₃.₂Dic₃ C₃³:Dic₃ He₃.₃Dic₃ He₃:Dic₃ C₃³:₄C₁₂ He₃.₄Dic₃ C₃²:CSU₂(F₃) C₆²:₅Dic₃ C₆.(S₃xA₄) C₆²:₄C₁₂
C₃²:C₁₂ is a maximal quotient of
He₃:₃C₈ C₃²:C₃₆ C₃²:Dic₉ He₃:C₁₂ C₃³:C₁₂ He₃.C₁₂ He₃.Dic₃ He₃.₂C₁₂ He₃.₂Dic₃ C₃³:₄C₁₂ C₆²:₅Dic₃ C₆²:₄C₁₂

Matrix representation of C₃²:C₁₂ ►in GL₆(F₁₃)

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

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

12	7	12	7	7	7
6	1	6	1	1	6
7	12	12	7	7	12
6	6	6	1	6	6
7	12	7	7	7	7
6	6	1	6	1	6

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

C₃²:C₁₂ in GAP, Magma, Sage, TeX

C_3^2\rtimes C_{12}

% in TeX

G:=Group("C3^2:C12");

// GroupNames label

G:=SmallGroup(108,8);

// by ID

G=gap.SmallGroup(108,8);

# by ID

G:=PCGroup([5,-2,-3,-2,-3,-3,30,483,488,1804]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃²:C₁₂ in TeX
Character table of C₃²:C₁₂ in TeX

G = C32:C12 order 108 = 22·33

The semidirect product of C32 and C12 acting via C12/C2=C6

G = C₃²:C₁₂ order 108 = 2²·3³

The semidirect product of C₃² and C₁₂ acting via C₁₂/C₂=C₆