C4^2:D6 - GroupNames

G = C₄²⋊D₆ order 192 = 2⁶·3

The semidirect product of C₄² and D₆ acting faithfully

non-abelian, soluble, monomial, rational

Aliases: C₄²⋊D₆, C₂³.₃S₄, C₄⋊₁D₄⋊S₃, C₄²⋊S₃⋊C₂, C₄²⋊C₃⋊₁C₂², C₂².₃(C₂×S₄), C₂³.A₄⋊₂C₂, SmallGroup(192,956)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄² — C₄²⋊C₃ — C₄²⋊D₆

Chief series

C₁ — C₂² — C₄² — C₄²⋊C₃ — C₄²⋊S₃ — C₄²⋊D₆

Lower central

C₄²⋊C₃ — C₄²⋊D₆

Upper central

C₁

Generators and relations for C₄²⋊D₆
G = < a,b,c,d | a⁴=b⁴=c⁶=d²=1, dad=cbc^-1=ab=ba, cac^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 480 in 83 conjugacy classes, 10 normal (8 characteristic)
C₁, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, D₄, Q₈, C₂³, C₂³, A₄, D₆, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, S₄, C₂×A₄, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₄²⋊C₃, C₂×S₄, D₄⋊₄D₄, C₄²⋊S₃, C₂³.A₄, C₄²⋊D₆
Quotients: C₁, C₂, C₂², S₃, D₆, S₄, C₂×S₄, C₄²⋊D₆

Character table of C₄²⋊D₆

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	6	8A	8B
size	1	3	4	12	12	12	32	6	6	12	12	32	24	24
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	2	2	-2	0	-2	0	-1	2	2	0	0	1	0	0	orthogonal lifted from D₆
ρ₆	2	2	2	0	2	0	-1	2	2	0	0	-1	0	0	orthogonal lifted from S₃
ρ₇	3	3	-3	-1	1	1	0	-1	-1	1	-1	0	-1	1	orthogonal lifted from C₂×S₄
ρ₈	3	3	-3	1	1	-1	0	-1	-1	-1	1	0	1	-1	orthogonal lifted from C₂×S₄
ρ₉	3	3	3	-1	-1	-1	0	-1	-1	-1	-1	0	1	1	orthogonal lifted from S₄
ρ₁₀	3	3	3	1	-1	1	0	-1	-1	1	1	0	-1	-1	orthogonal lifted from S₄
ρ₁₁	6	-2	0	-2	0	0	0	-2	2	0	2	0	0	0	orthogonal faithful
ρ₁₂	6	-2	0	0	0	2	0	2	-2	-2	0	0	0	0	orthogonal faithful
ρ₁₃	6	-2	0	2	0	0	0	-2	2	0	-2	0	0	0	orthogonal faithful
ρ₁₄	6	-2	0	0	0	-2	0	2	-2	2	0	0	0	0	orthogonal faithful

Permutation representations of C₄²⋊D₆
►On 12 points - transitive group 12T112

Generators in S₁₂

(1 4 7 10)(2 5 8 11)

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

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

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

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

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

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

G:=TransitiveGroup(12,112);

►On 12 points - transitive group 12T113

Generators in S₁₂

(1 4 12 9)(2 5 7 10)

(1 9 12 4)(3 6 8 11)

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

(1 3)(4 6)(8 12)(9 11)

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

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

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

G:=TransitiveGroup(12,113);

►On 12 points - transitive group 12T114

Generators in S₁₂

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

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

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

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

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

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

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

G:=TransitiveGroup(12,114);

►On 12 points - transitive group 12T115

Generators in S₁₂

(1 12 5 9)(2 7 6 10)

(1 9 5 12)(3 8 4 11)

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

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

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

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

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

G:=TransitiveGroup(12,115);

►On 16 points - transitive group 16T431

Generators in S₁₆

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

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

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

(2 4)(5 8)(6 7)(9 10)(11 15)(12 14)

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

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

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

G:=TransitiveGroup(16,431);

►On 24 points - transitive group 24T530

Generators in S₂₄

(1 18 8 20)(2 13 9 21)(4 23 11 15)(5 24 12 16)

(1 20 8 18)(3 14 10 22)(4 15 11 23)(6 19 7 17)

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

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

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

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

G:=TransitiveGroup(24,530);

►On 24 points - transitive group 24T531

Generators in S₂₄

(2 5 13 16)(3 6 14 17)(7 19 22 10)(9 12 24 21)

(1 15 18 4)(2 16 13 5)(8 20 23 11)(9 21 24 12)

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

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

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

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

G:=TransitiveGroup(24,531);

►On 24 points - transitive group 24T532

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,532);

►On 24 points - transitive group 24T533

Generators in S₂₄

(1 18 8 20)(2 13 9 21)(4 23 11 15)(5 24 12 16)

(1 20 8 18)(3 14 10 22)(4 15 11 23)(6 19 7 17)

(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 3)(4 6)(7 11)(8 10)(14 18)(15 17)(19 23)(20 22)

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

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

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

G:=TransitiveGroup(24,533);

►On 24 points - transitive group 24T534

Generators in S₂₄

(2 14 9 17)(3 15 7 18)(4 20 10 23)(6 19 12 22)

(1 16 8 13)(2 17 9 14)(5 21 11 24)(6 22 12 19)

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

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

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

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

G:=TransitiveGroup(24,534);

►On 24 points - transitive group 24T535

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,535);

►On 24 points - transitive group 24T536

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,536);

►On 24 points - transitive group 24T537

Generators in S₂₄

(1 18 8 20)(2 13 9 21)(4 23 11 15)(5 24 12 16)

(1 20 8 18)(3 14 10 22)(4 15 11 23)(6 19 7 17)

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

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

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

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

G:=TransitiveGroup(24,537);

►On 24 points - transitive group 24T538

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,538);

►On 24 points - transitive group 24T539

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,539);

►On 24 points - transitive group 24T540

Generators in S₂₄

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

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

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

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

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

G:=TransitiveGroup(24,540);

►On 24 points - transitive group 24T541

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,541);

Polynomial with Galois group C₄²⋊D₆ over ℚ

action	f(x)	Disc(f)
12T112	x¹²-18x⁸-8x⁴+16	2⁷²·29⁸
12T113	x¹²+4x¹⁰-3x⁸-6x⁶+13x⁴-12x²+4	2⁴⁰·79⁴
12T114	x¹²-13x¹⁰+61x⁸-125x⁶+105x⁴-27x²+2	2²⁵·733⁴
12T115	x¹²-2x⁸+3x⁴-4	-2⁴²·5⁸

Matrix representation of C₄²⋊D₆ ►in GL₆(ℤ)

0	1	0	0	0	0
-1	0	0	0	0	0
0	0	0	-1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	-1	0	0	0	0
1	0	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	-1	0

0	0	0	0	0	-1
0	0	0	0	-1	0
0	-1	0	0	0	0
-1	0	0	0	0	0
0	0	0	-1	0	0
0	0	-1	0	0	0

0	0	0	0	-1	0
0	0	0	0	0	-1
0	0	-1	0	0	0
0	0	0	-1	0	0
-1	0	0	0	0	0
0	-1	0	0	0	0

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,-1,0,0,0,0,-1,0,0,0,0,0],[0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0] >;

C₄²⋊D₆ in GAP, Magma, Sage, TeX

C_4^2\rtimes D_6

% in TeX

G:=Group("C4^2:D6");

// GroupNames label

G:=SmallGroup(192,956);

// by ID

G=gap.SmallGroup(192,956);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,170,675,2194,185,360,424,1271,1173,102,6053,1027,1784]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,d*a*d=c*b*c^-1=a*b=b*a,c*a*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

Export

Character table of C₄²⋊D₆ in TeX

G = C42⋊D6 order 192 = 26·3

The semidirect product of C42 and D6 acting faithfully

G = C₄²⋊D₆ order 192 = 2⁶·3

The semidirect product of C₄² and D₆ acting faithfully