C3^3:D4 - GroupNames

G = C₃³⋊D₄ order 216 = 2³·3³

2^nd semidirect product of C₃³ and D₄ acting faithfully

Aliases: C₃³⋊₂D₄, S₃²⋊S₃, C₃⋊₂S₃≀C₂, C₃⋊S₃.₁D₆, C₃³⋊C₄⋊₂C₂, C₃²⋊₄D₆⋊₁C₂, C₃²⋊₃(C₃⋊D₄), (C₃×S₃²)⋊₂C₂, (C₃×C₃⋊S₃).₄C₂², SmallGroup(216,158)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃×C₃⋊S₃ — C₃³⋊D₄

Chief series

C₁ — C₃ — C₃³ — C₃×C₃⋊S₃ — C₃²⋊₄D₆ — C₃³⋊D₄

Lower central

C₃³ — C₃×C₃⋊S₃ — C₃³⋊D₄

Upper central

C₁

Generators and relations for C₃³⋊D₄
G = < a,b,c,d,e | a³=b³=c³=d⁴=e²=1, ab=ba, ac=ca, dad^-1=b^-1, eae=b, bc=cb, dbd^-1=ebe=a, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 384 in 60 conjugacy classes, 11 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², S₃, C₆, D₄, C₃², C₃², Dic₃, D₆, C₂×C₆, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃⋊D₄, C₃³, C₃²⋊C₄, S₃², S₃², S₃×C₆, S₃×C₃², C₃×C₃⋊S₃, C₃×C₃⋊S₃, S₃≀C₂, C₃³⋊C₄, C₃×S₃², C₃²⋊₄D₆, C₃³⋊D₄
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, C₃⋊D₄, S₃≀C₂, C₃³⋊D₄

Character table of C₃³⋊D₄

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	4	6A	6B	6C	6D	6E	6F	6G
size	1	6	9	18	2	4	4	4	4	8	54	6	6	12	12	12	18	36
ρ₁	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	linear of order 2
ρ₃	1	-1	1	-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	-1	1	1	linear of order 2
ρ₅	2	0	-2	0	2	2	2	2	2	2	0	0	0	0	0	0	-2	0	orthogonal lifted from D₄
ρ₆	2	-2	2	0	-1	2	-1	-1	2	-1	0	1	1	1	1	-2	-1	0	orthogonal lifted from D₆
ρ₇	2	2	2	0	-1	2	-1	-1	2	-1	0	-1	-1	-1	-1	2	-1	0	orthogonal lifted from S₃
ρ₈	2	0	-2	0	-1	2	-1	-1	2	-1	0	-√-3	√-3	√-3	-√-3	0	1	0	complex lifted from C₃⋊D₄
ρ₉	2	0	-2	0	-1	2	-1	-1	2	-1	0	√-3	-√-3	-√-3	√-3	0	1	0	complex lifted from C₃⋊D₄
ρ₁₀	4	2	0	0	4	1	1	1	-2	-2	0	2	2	-1	-1	-1	0	0	orthogonal lifted from S₃≀C₂
ρ₁₁	4	-2	0	0	4	1	1	1	-2	-2	0	-2	-2	1	1	1	0	0	orthogonal lifted from S₃≀C₂
ρ₁₂	4	0	0	-2	4	-2	-2	-2	1	1	0	0	0	0	0	0	0	1	orthogonal lifted from S₃≀C₂
ρ₁₃	4	0	0	2	4	-2	-2	-2	1	1	0	0	0	0	0	0	0	-1	orthogonal lifted from S₃≀C₂
ρ₁₄	4	2	0	0	-2	1	^-1-3√-3/₂	^-1+3√-3/₂	-2	1	0	-1+√-3	-1-√-3	ζ₆	ζ₆⁵	-1	0	0	complex faithful
ρ₁₅	4	-2	0	0	-2	1	^-1+3√-3/₂	^-1-3√-3/₂	-2	1	0	1+√-3	1-√-3	ζ₃	ζ₃²	1	0	0	complex faithful
ρ₁₆	4	2	0	0	-2	1	^-1+3√-3/₂	^-1-3√-3/₂	-2	1	0	-1-√-3	-1+√-3	ζ₆⁵	ζ₆	-1	0	0	complex faithful
ρ₁₇	4	-2	0	0	-2	1	^-1-3√-3/₂	^-1+3√-3/₂	-2	1	0	1-√-3	1+√-3	ζ₃²	ζ₃	1	0	0	complex faithful
ρ₁₈	8	0	0	0	-4	-4	2	2	2	-1	0	0	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₃³⋊D₄
►On 12 points - transitive group 12T116

Generators in S₁₂

(2 10 5)(4 7 12)

(1 8 9)(3 11 6)

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

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

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

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

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

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

G:=TransitiveGroup(12,116);

►On 12 points - transitive group 12T120

Generators in S₁₂

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

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

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

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

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

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

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

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

G:=TransitiveGroup(12,120);

►On 18 points - transitive group 18T103

Generators in S₁₈

(1 18 16)(4 8 10)(6 13 11)

(2 15 17)(3 9 7)(5 14 12)

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

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

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

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

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

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

G:=TransitiveGroup(18,103);

►On 18 points - transitive group 18T106

Generators in S₁₈

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

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

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

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

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

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

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

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

G:=TransitiveGroup(18,106);

►On 24 points - transitive group 24T556

Generators in S₂₄

(1 20 21)(3 23 18)(5 14 10)(7 12 16)

(2 17 22)(4 24 19)(6 15 11)(8 9 13)

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

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

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

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

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

G:=TransitiveGroup(24,556);

►On 24 points - transitive group 24T560

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,560);

►On 27 points - transitive group 27T81

Generators in S₂₇

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

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

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

(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)

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

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

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

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

G:=TransitiveGroup(27,81);

C₃³⋊D₄ is a maximal subgroup of S₃×S₃≀C₂
C₃³⋊D₄ is a maximal quotient of C₃⋊S₃.₂D₁₂ S₃²⋊Dic₃ C₃³⋊C₄⋊C₄ C₃³⋊D₈ C₃³⋊₆SD₁₆ C₃³⋊₇SD₁₆ C₃³⋊Q₁₆

Polynomial with Galois group C₃³⋊D₄ over ℚ

action	f(x)	Disc(f)
12T116	x¹²-3x⁹+3x⁶-3x³+3	3²³·7³
12T120	x¹²-2x⁹-12x⁶-8x³-2	-2²⁰·3²¹·7⁶

Matrix representation of C₃³⋊D₄ ►in GL₄(𝔽₇) generated by

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

6	2	1	1
2	6	6	1
0	0	1	0
0	0	0	2

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

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

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

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

C₃³⋊D₄ in GAP, Magma, Sage, TeX

C_3^3\rtimes D_4

% in TeX

G:=Group("C3^3:D4");

// GroupNames label

G:=SmallGroup(216,158);

// by ID

G=gap.SmallGroup(216,158);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,579,201,111,244,376,5189]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=b^-1,e*a*e=b,b*c=c*b,d*b*d^-1=e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

Export

Character table of C₃³⋊D₄ in TeX

G = C33⋊D4 order 216 = 23·33

2nd semidirect product of C33 and D4 acting faithfully

G = C₃³⋊D₄ order 216 = 2³·3³

2^nd semidirect product of C₃³ and D₄ acting faithfully